Computational Mathematics and Applications [1 ed.] 9789811584978, 9789811584985

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Forum for Interdisciplinary Mathematics

Dia Zeidan Seshadev Padhi Aliaa Burqan Peer Ueberholz   Editors

Computational Mathematics and Applications

Forum for Interdisciplinary Mathematics Editors-in-Chief Viswanath Ramakrishna, University of Texas, Texas, USA Zhonghai Ding, University of Nevada, Las Vegas, USA Editorial Board Ravindra B. Bapat, Indian Statistical Institute, Delhi, India Balasubramaniam Jayaram, Indian Institute of Technology Hyderabad, India Ashis Sengupta, Indian Statistical Institute, Kolkata, India Bhu Dev Sharma, Jaypee Institute of Information Technology, Noida, India P. V. Subrahmanyam, Indian Institute of Technology Madras, India

The Forum for Interdisciplinary Mathematics is a Scopus-indexed book series. It publishes high-quality textbooks, monographs, contributed volumes and lecture notes in mathematics and interdisciplinary areas where mathematics plays a fundamental role, such as statistics, operations research, computer science, financial mathematics, industrial mathematics, and bio-mathematics. It reflects the increasing demand of researchers working at the interface between mathematics and other scientific disciplines.

More information about this series at http://www.springer.com/series/13386

Dia Zeidan · Seshadev Padhi · Aliaa Burqan · Peer Ueberholz Editors

Computational Mathematics and Applications

Editors Dia Zeidan School of Basic Sciences and Humanities German Jordanian University Amman, Jordan Aliaa Burqan Department of Mathematics Zarqa University Zarqa, Jordan

Seshadev Padhi Department of Mathematics Birla Institute of Technology Mesra, Ranchi, Jharkhand, India Peer Ueberholz Institut für Modellbildung und Hochleistungsrechnen Hochschule Niederrhein Krefeld, Germany

ISSN 2364-6748 ISSN 2364-6756 (electronic) Forum for Interdisciplinary Mathematics ISBN 978-981-15-8497-8 ISBN 978-981-15-8498-5 (eBook) https://doi.org/10.1007/978-981-15-8498-5 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

Research topics related to mathematical sciences play an increasingly important role in building advanced scientific disciplines in rapid and cost-effective ways. The motivation to write this book came during the preparation of the 6th International Arab Conference on Mathematics and Computations (IACMC 2019), held at Zarqa University, Zarqa, Jordan. It grew not only from the international collaboration between the authors but rather from the long need for a research-based book from different parts of the world for researchers and professionals working in the fields of computational and applied mathematics. This book constitutes the first attempt in Jordanian literature to scientifically consider the extensive need of research development at the national and international levels with which mathematics deals. The aim is to provide brief and reliably expressed research topics that will enable those new to or not aware of mathematical sciences in this part of the world. This supports the active placement of personnel into the international community and replacement throughout collaborative networking. Further, all editors wanted to create a book that would comprise many of the recent research results that are available in the world but are simply not in books. While the book has not been precisely planned to address any branch of mathematics, it presents contributions of the relevant topics to do so. The topics chosen for the book are those that we have found of significant interest to many researchers in the world. These also are topics that are applicable in many fields of computational and applied mathematics. As a lead editor of this book, it has been my pleasure to work with the co-editors, authors and reviewers as they cooperated well beyond expectations. The editors also would like to thank the Senior Publishing Editor at Springer Nature, Shamim Ahmad, for supporting the idea of this book. We also gratefully acknowledge the German Jordanian University, Jordan; Birla Institute of Technology, India; Zarqa University, Jordan; and Niederrhein University of Applied Sciences, Germany, for their kind cooperation in supporting us to publish this book. Amman, Jordan

Dia Zeidan

v

Contents

Analysing Differential Equations with Uncertainties via the Liouville-Gibbs Theorem: Theory and Applications . . . . . . . . . . . . V. Bevia, C. Burgos, J.-C. Cortés, A. Navarro-Quiles, and R.-J. Villanueva Solving Time-Space-Fractional Cauchy Problem with Constant Coefficients by Finite-Difference Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reem Edwan, Rania Saadeh, Samir Hadid, Mohammed Al-Smadi, and Shaher Momani On Modification of an Adaptive Stochastic Mirror Descent Algorithm for Convex Optimization Problems with Functional Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mohammad S. Alkousa

1

25

47

Inductive Description of Quadratic Lie and Pseudo-Euclidean Jordan Triple Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Amir Baklouti and Samiha Hidri

65

Comparative Study of Some Numerical Methods for the Standard FitzHugh-Nagumo Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Koffi Messan Agbavon, Appanah Rao Appadu, and Bilge ˙Inan

95

Analytical Solution of Neutron Diffusion Equation in Reflected Reactors Using Modified Differential Transform Method . . . . . . . . . . . . . . 129 Mohammed Shqair and Essam R. El-Zahar Second-Order Perturbed State-Dependent Sweeping Process with Subsmooth Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 Doria Affane and Mustapha Fateh Yarou Membrane Hydrogen Mixture Separation: Modelling and Analysis . . . . 171 Khaled Alhussan, Kirill Delendik, Natalia Kolyago, Oleg Penyazkov, and Olga Voitik

vii

viii

Contents

An Efficient Hybrid Conjugate Gradient Method for Unconstrained Optimisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 Talat Alkhouli, Hatem S. A. Hamatta, Mustafa Mamat, and Mohd Rivaie On the Polynomial Decay of the Wave Equation with Wentzell Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 Karima Laoubi and Djamila Seba Solutions of Fractional Verhulst Model by Modified Analytical and Numerical Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 Shatha Hasan, Samir Hadid, Mohammed Al-Smadi, Omar Abu Arqub, and Shaher Momani Is It Worthwhile Considering Orthogonality in Generalised Polynomial Chaos Expansions Applied to Solving Stochastic Models? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 Julia Calatayud, J.-C. Cortés, Marc Jornet, and Laura Villafuerte

About the Authors

Dia Zeidan is Associate Professor of Applied and Computational Mathematics at the German Jordanian University, Amman, Jordan, and Elected Fellow of the European Society of Computational Methods in Sciences and Engineering (ESCMSE). Although he is based in a developing country, he has been an active researcher in developing mathematical and numerical tools of multiphase fluid flow problems for several years. He is recognized for research contributions in applied and computational mathematics with multiphase flows including his creative approaches to teaching and research. His work has been highly interdisciplinary, involving international collaborations with applied and computational researchers. Among various institutional obligations, he has been a visitor of several important international research groups bridging with national research infrastructure gaps in Jordan. He serves on several expert review panels, as a technical editor and reviewer for several peer-review journals and as a member of several programme committees of technical conferences around the world. Seshadev Padhi is Professor and Head of the Department of Mathematics, Birla Institute of Technology, Mesra, Jharkhand, India. He received his Ph.D. in the topic of “Oscillation theory of third order differential equations”. Among the many awards, he received is the BOYSCAST (Better Opportunities for Young Scientists in Chosen Areas of Science and Technology) fellow by the Department of Science and Technology (DST), Government of India, in 2004, to visit Mississippi State University, USA. Subsequently, Prof. Padhi did his postdoctoral research work in Mississippi State University, USA. In addition, he has visited several institutes of international repute such as Florida Institute of Technology, Melbourne, Florida USA, to work in collaboration with Prof. T. Gnanabhakar in 2006, Texas State University at San Marcos, Texas, USA, to work in collaboration with Professor Julio G. Dix in 2009, University of Tennessee at Chattanooga, Chattanooga, Tennessee, USA, during 2011, 2012 and 2013 to work in collaboration with Prof. John R. Graef and University of Szeged, Szeged, Hungary, in 2007 and 2011, to work in collaboration with Prof. Tibor Krisztin. Furthermore, he visited ETH, Zurich, Switzerland, under the Borel Set Theory Programme in 2005 and also several countries to deliver lectures in different international conferences and workshops. In 2003, he received the UNESCO ix

x

About the Authors

travel and lodging grant to visit ICTP, Trieste, Italy. He has more than 150 research papers in reputed international journals and conferences and has written two books in mathematics where you will find more information on his biography in the field. Aliaa Burqan is Associate Professor and Dean of the Faculty of Science at Zarqa University, Zarqa, Jordan. She holds a Ph.D. in Mathematics from the University of Jordan, Amman, Jordan. Her areas of research interests are functional analysis, matrix analysis and operator theory. She has several research papers published in reputed international journals and conferences. She is an active member in the organizing committee of all Mathematical Conferences taking a place at Zarqa University from 2006 up to now and the Chairman of the last one, IACMC 2019. She is also serving as scientific and advisory board member of some educational bodies. During the last few years, she has supervised several students who are continuing their postgraduate studies across the world. Peer Ueberholz is Professor of Parallel Computing at the Department of Electrical Engineering and Computer Science in the Niederrhein University of Applied Science, Krefeld, Germany. He is a founding member of the interdisciplinary Institute of Modelling and High-Performance Computing at the Niederrhein University. He also has studies in theoretical physics and received his diploma from Bochum University in quantum field theory in 1985 and his doctorate degree from Wuppertal University in 1989 in lattice gauge theory. He is recognized for his research contributions in applied and computational physics, computer science and engineering. His current research interest is in the field of high-performance computing with applications to computational fluid dynamics and machine learning. He serves on several expert review panels and as a reviewer for several peer-review journals and conference. He also published over seventy peer-reviewed journal and conference papers with h-index of 19 and more than 1500 citations according to Google Scholar. Several researchers across the globe have cited his published work.

Analysing Differential Equations with Uncertainties via the Liouville-Gibbs Theorem: Theory and Applications V. Bevia, C. Burgos, J.-C. Cortés, A. Navarro-Quiles, and R.-J. Villanueva

Abstract In this contribution, we revisit the Liouville-Gibbs theorem for dynamical systems. This theorem states that a partial differential equation that is satisfied by the probability density function of the solution stochastic process of an initial value problem with uncertainties in its initial condition, forcing term and coefficients. We show its key role in the setting of dynamical systems with uncertainties by means of a variety of illustrative models appearing in several scientific realms that include physics and biology. Specifically, we deal with the undamped and damped linear oscillator, and the logistic model. These models are formulated via random differential equations with a finite degree of randomness. Numerical simulations and computations are carried out to illustrate the capability of the Liouville-Gibbs theorem.

1 Introduction and Preliminaries The key role played by differential equations is mainly justified by their wide range of applications in many scientific fields including Physics, Chemistry, Engineering, Biology, Epidemiology, Economics, etc. In practice, model inputs (coefficients, source terms and initial/boundary conditions) appearing in the formulation of differential equations often need to be set from sampling, experimental measurements or metadata excerpted from the extant literature. This approach entails that model inputs involve uncertainties and then they must be treated as random variables (r.v.’s) or stochastic processes (s.p.’s) rather than deterministic constants or functions, respectively. This approach leads to formulate random differential equations (r.d.e.’s) in mathematical modelling [1–3]. Apart from studying relevant theoretical questions, V. Bevia · C. Burgos · J.-C. Cortés (B) · A. Navarro-Quiles · R.-J. Villanueva Instituto de Matemática Multidisciplinar, Universitat Politècnica de València, Camino de Vera, s/n, 46022 Valencia, Spain e-mail: [email protected] URL: https://www.imm.upv.es © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 D. Zeidan et al. (eds.), Computational Mathematics and Applications, Forum for Interdisciplinary Mathematics, https://doi.org/10.1007/978-981-15-8498-5_1

1

2

V. Bevia et al.

like existence and uniqueness conditions to initial value problems (i.v.p.’s), as well as to devise methods for computing their solutions, say X (t) = X (t)(ω) (ω ∈ , being (, F, P) a complete probability space), in the setting of r.d.e.’s a primary objective is to determine the main statistical functions of the solution s.p. The most important information in this latter regard includes determining the mean (E[X (t)]) and the variance (V[X (t)]), since they provide the average and the variability of the solution, and moreover, they allow to construct pointwise and probabilistic predictions. However, a more complete goal consists of computing the first probability density function (1-p.d.f.), say f (t, x), since from this deterministic function one can calculate the one-dimensional moments of arbitrary order  μk = E[(X (t)) ] = k

+∞

−∞

xk f (t, x) dx, k = 1, 2, . . . ,

(1)

provided they exist. Observe that, in particular, E[X (t)] = μ1 and V[X (t)] = μ2 − μ21 . Besides, the computation of the 1-p.d.f. enables the calculation that at any specific time instant, say ˆt , the solution lies within a specific interval of interest, say [a, b], via its direct integration P[{ω ∈  : a ≤ X (ˆt )(ω) ≤ b}] =



b

f (ˆt , x) dx.

a

In the context of r.d.e.’s, the main approach to determine the 1-p.d.f. of the solution of dynamical systems with randomness is the random variable transformation (r.v.t.) technique. This method has been successfully applied in dealing with random difference equations [4, 5], random o.d.e.’s [6, 7] and random partial differential equations [8, 9]. As it can be checked in these papers, the successful application of r.v.t. method relies heavily on defining an appropriate invertible mapping based on the knowledge of an explicit expression of the solution s.p. and then computing its Jacobian. In this contribution, we study an alternative approach, based on the Liouville-Gibbs partial differential equation (p.d.e.) for dynamical systems, in order to determine the 1-p.d.f. of r.d.e.’s. The main advantage of this approach, with respect to the r.v.t. method, is that it provides an explicit expression of the 1-p.d.f. in terms of the data avoiding the search of any ad hoc invertible mapping as well as the computation of its Jacobian. For the sake of completeness, hereinafter we introduce some definitions, notations and results that will be required throughout this contribution (see [1, Chap. 4], [10, 11] for further details). Let (, F, P) be a complete probability space. We will work in the Hilbert space (L2 (), , ), whose elements are real r.v.’s X :  → R and  the inner product is defined by X , Y  = E[XY ], X , Y ∈ L2 (), being E[·] =  · dP the expectation operator. From this inner product, one derives the corresponding norm X 2 = (E[X 2 ])1/2 . Elements in space L2 () are termed second-order r.v.’s (2-r.v.’s) and they have finite variance (V[X ] = E[X 2 ] − (E[X ])2 < ∞, since the finiteness of E[X 2 ] entails E[X ] < ∞ too). The convergence associated to  · 2 – norm is referred to as mean square (m.s.) convergence and it is defined as follows. Let {Xn : n ≥ 0} be a sequence of r.v.’s in L2 () and X ∈ L2 (), we say that Xn is

Analysing Differential Equations with Uncertainties …

3

m.s. convergent to X if and only if Xn − X 2 → 0 as n → ∞. This fact is denoted ·2

by Xn −−−→ X . This stochastic convergence has the following distinctive property n→∞ [1, Theorem 4.3.1]: ·2

Xn −−−→ X ⇒ E[Xn ] −−−→ E[X ] and V[Xn ] −−−→ V[X ]. n→∞

n→∞

n→∞

(2)

A random function (or s.p.), say X (t) ≡ {X (t) : t ∈ T ⊂ R}, defined in the space L2 () is such that X (t) ∈ L2 () for each t ∈ T , i.e. X (t) is a 2-r.v., and it is called a second-order s.p. (2-s.p.). The concept of m.s. derivative of a 2-s.p., X˙ (t), is defined in terms of m.s. convergence via the incremental quotient     X (t + h) − X (t)  ˙ − X (t) lim   = 0. h→0 h 2 As a consequence of property (2) and the fact that m.s. derivatives are defined in terms of m.s. limits, one gets that the expectation operator and the m.s. derivative commute [1, p. 97], i.e. d E[X˙ (t)] = (3) (E[X (t)]) . dt The previous scalar concepts can be straightforwardly extended to the multidimensional scenario leading to the Banach space of second-order random vectors, (L2n (),  · n ), where   L2n () := X = (X1 , . . . , Xn )T : Xj ∈ L2 (), 1 ≤ j ≤ n , Xn := max Xj 2 . 1≤j≤n

In this contribution, we will take advantage of the Liouville-Gibbs p.d.e. to obtain the 1-p.d.f. of solution s.p. to random initial value problems (i.v.p.’s) of the form 

˙ X(t) = g(t, X(t)), t ≥ t0 , X(t0 ) = X0 ,

(4)

where g : [t0 , ∞[×L2n () → L2n () is a continuously differentiable function, X0 is an absolutely continuous random vector in L2n (), X(t) is a 2-s.p. defined in L2n (), ˙ denotes the m.s. derivative with respect to time. As it will be seen later, the and X(t) Liouville-Gibbs p.d.e. will be set combining the characteristic function of a s.p. and the deterministic Fourier transform. For the sake of clarity in the subsequent presentation, we recall some definitions and properties related to the Fourier transform that will be required later.Hereinafter, L1 (Rn ) stands for the set of absolutely integrable functions on Rn , i.e. Rn |f (x)| dx < ∞, and F[f (x)](v) =

1 (2π)n

 Rn

T

eiv x f (x) dx,

4

V. Bevia et al.

√ is its Fourier transform, where i = −1 denotes the imaginary unit and v = (v1 , . . . , vn )T and x = (x1 , . . . , xn )T are column vectors in Rn . Theorem 1 ([12, Prop. 17.2.1 (ii)]) Let f ∈ L1 (Rn ) be continuously differentiable ∂f ∈ L1 (Rn ) and lim|xj |→∞ f (x) = 0, then with respect to xj , and assume that ∂x j  F

∂f (x) (v) = 2πivj F[f (x)](v), j = 1, . . . , n. ∂xj

(5)

If we consider the change of variable: u = −2πv, property (5) can be written as  F

∂f (x) (u) = −iuj F[f (x)](u). ∂xj

(6)

The following theorem is known as the inversion Fourier identity. Theorem 2 ([12, Theorem 18.1.1]) If f , F[f ] ∈ L1 (Rn ) and  F[f (v)](x) =

T

Rn

e2πix v f (v)dv,

then F[F[f ]](x) = f (x), almost everywhere, particularly, in every x ∈ Rn where f is continuous. This contribution is organised as follows. In Sect. 2, we revisit the classical Liouville theorem for dynamical systems focusing on its adaptation to the context of r.d.e.’s. In the deterministic context, this result establishes that the density of the solution of a dynamical system is an integral invariant of the motion and satisfies a p.d.e., called Liouville-Gibbs equation (also termed continuity equation in the context of Hydrodynamics) [13, 14]. In the probabilistic setting, we will see that this result can be interpreted as the p.d.e. satisfied for the 1-p.d.f. of the solution s.p of an r.d.e. Specifically, in Sect. 2.1, we will derive the Liouville-Gibbs p.d.e. in the context of random dynamical systems of type (4), and then, in Sect. 2.2 we will obtain an explicit expression of its solution, which represents the 1-p.d.f. of the solution s.p. to the random i.v.p. (4). Section 3 is devoted to studying several randomised models, formulated via differential equations, which appear in different scientific fields. By taking advantage of results exhibited in Sect. 2, we will obtain explicit expressions of the 1-p.d.f. of their corresponding solution s.p.’s. Then, we will carry out some numerical simulations to illustrate the main theoretical results studied in previous sections. Conclusions are drawn in Sect. 4.

Analysing Differential Equations with Uncertainties …

5

2 The 1-P.D.F. of a Random System and the Liouville-Gibbs Equation Throughout this section, we will develop the main theoretical ideas and results of this contribution, first introduced by [1], with great detail. In the first subsection, we will see the relationship between the 1-p.d.f. of the solution s.p. of a random i.v.p. and the Liouville-Gibbs, or continuity, equation. Afterwards, in the following subsection, we will see when we can guarantee a solution and how to obtain it.

2.1 The Liouville-Gibbs Partial Differential Equation In this subsection, we shall show that a p.d.f., f = f (t, x), associated to the solution s.p. of the random i.v.p. (4) satisfies the following p.d.e.:

∂(f gj ) ∂f + = 0, ∂t ∂xj j=1 n

(7)

where gj = gj (x1 , . . . , xn ), 1 ≤ j ≤ n, denotes the components of mapping g defining the right-hand side of the r.d.e. in (4). Equation (7) is called the Liouville-Gibbs p.d.e. In order to derive this equation, let us fix t ∈ [t0 , ∞[ and consider the definition of the characteristic function of the random vector X(t) = (X1 (t), . . . , Xn (t))T (which corresponds to the evaluation of the solution s.p. to i.v.p. (4) at the time instant t ∈ [t0 , ∞[),  T  T φ(t, u) = E eiu X(t) = eiu x f (t, x) dx = (2π)n F[f (t, x)](u), (8) Rn

where u = (u1 , . . . , un )T ∈ Rn and F[·] is the Fourier transform operator. Now, we differentiate expression (8) with respect to t and apply the commutation between the m.s. derivative and the expectation operator (see (3)). This yields    n ∂  n ∂ ∂φ(t, u) = E ei k=1 uk Xk (t) = E ei k=1 uk Xk (t) ∂t ∂t ∂t  n  n



T iuT X(t) ˙ =E i uk Xk (t)e uk E[X˙ k (t)eiu X(t) ] =i k=1

=i

n

k=1

uk E[gk (t, X(t))eiu

k=1

T

X(t)

]=

n

k=1

 iuk

T

Rn

eiu x gk (t, x)f (t, x) dx.

(9)

6

V. Bevia et al.

On the one hand, according to (6) and (8), each addend in the last sum can be expressed in terms of the Fourier transform,  iuk

e Rn

iuT x

 ∂ gk (t, x)f (t, x) dx = (2π) F − (gk (t, x)f (t, x)) (u). ∂xk n

Then using this latter representation together with the linearity of the Fourier transform operator, expression (9) can be written as  n 

∂ ∂φ(t, u) n = (2π) F − (gk (t, x)f (t, x)) (u). ∂t ∂xk

(10)

k=1

On the other hand, if we directly differentiate (8) under the integral sign with respect to t, one gets ∂φ(t, u) = ∂t

 e

iuT x ∂f

Rn

 (t, x) ∂f (t, x) n dx = (2π) F (u). ∂t ∂t

(11)

Finally, subtracting (10) and (11) and using Theorem 2, i.e. the inversion Fourier transformation, one obtains the Liouville-Gibbs equation ∂f (t, x) ∂ + (gk (t, x)f (t, x)) = 0. ∂t ∂xk n

(12)

k=1

Remark 1 Notice that if we differentiate the product in (12), the Liouville-Gibbs equation can be expressed in terms of the divergence operator of map equivalently k ping g, i.e. ∇x · g = nk=1 ∂g , leading to ∂xk ∂f (t, x) ∂f (t, x)

+ gk (t, x) + f (t, x)∇x · g(t, x) = 0. ∂t ∂xk n

k=1

2.2 Solving the Liouville-Gibbs Equation In the context of applications, it is assumed that the p.d.f., f0 (x), of the absolutely continuous random initial condition (i.c.), X0 , of random i.v.p., (4), is known after sampling or experimental measurements. So, our goal is to determine the solution of the following i.v.p. for the Liouville-Gibbs p.d.e., ⎧ n

∂f (t, x) ⎪ ⎨ ∂f (t, x) + gk (t, x) = −f (t, x)∇x ·g(t, x), t > t0 , x ∈ Rn , ∂t ∂xk k=1 ⎪ ⎩ f (t0 , x) = f0 (x), x ∈ S0 ,

(13)

Analysing Differential Equations with Uncertainties …

7

where S0 denotes the interior of the support of f0 , which is assumed to be a C 1 hypersurface. To prove that there exists a unique solution to the i.v.p. (13), we will use the following theorem. Theorem 3 ([15, Theorem 1.10]) Let S be a C 1 hypersurface in Rn . Consider the following i.v.p. ⎧ n

∂u ⎪ ⎨ ak (x, u) (x) = b(x, u), x ∈ Rn , ∂xk (14) k=1 ⎪ ⎩ u(x) = ψ(x), x ∈ S, where ak , b and ψ are C 1 real-valued functions. Additionally, suppose that the vector (a1 (x, ψ(x)), . . . , an (x, ψ(x))) is not tangent to S at any point. Then, there exists a neighbourhood  of S in Rn such that there exists a unique solution u ∈ C 1 () of the i.v.p. (14). Let us check that i.v.p. (13) verifies the hypotheses of Theorem 3. We define S := {(t0 , x) : x ∈ S0 }, which is a smooth hypersurface parametrically given by σ : S0 −→ Rn+1 , σ(s) := (σ1 (s), . . . , σn+1 (s)) = (t0 , s), where s = (s1 , . . . , sn ) ∈ S0 ⊂ Rn . Let us see that (1, g1 (t0 , f0 (x)), . . . , gn (t0 , f0 (x))) is not tangent to S at any (t0 , x) ∈ S. Using the previous parametrization, we obtain ⎡ ∂σ (s) 1 1 ⎢ ∂σ∂s2 (s) ⎢ ⎢ ∂s1 det ⎢ .. ⎢ . ⎣

···

∂σ1 (s) ∂sn ∂σ2 (s) ∂sn

1

⎤

⎥ g1 (t0 , f0 (s))⎥ ⎥ ⎥ .. .. ⎥ . . ⎦ ∂σn+1 (s) ∂σn+1 (s) · · · g (t , f (s)) n 0 0 ∂s1 ∂sn ⎡ ⎤ 0 ··· 0 1 ⎢ g1 (t0 , f0 (s)) ⎥ ⎢ ⎥ = det ⎢ ⎥ = (−1)n+1  = 0, ∀s ∈ S0 , .. ⎣ In ⎦ . gn (t0 , f0 (s)) ··· .. .

where In denotes the identity matrix of size n. Therefore, i.v.p. (13) verifies the hypothesis of Theorem 3 and we can guarantee the local existence and uniqueness of solution for the Liouville-Gibbs i.v.p. (13). To calculate its solution, first we apply the Lagrange-Charpit technique [16], which leads to the following set of differential equations: df dx1 dxn dt = = = ··· = . 1 −f ∇x · g g1 gn

(15)

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V. Bevia et al.

Secondly, we apply the method of characteristics [17] for the above equations involving the variables t, x1 , . . . , xn . This allows us to establish the following set of differential equations formulated in terms of the auxiliary variables s and r = (r1 , . . . , rn ): dt (s, r) = 1, t(0, r) = 0, ds dx1 (s, r) = g1 (t, x), x1 (0, r) = r1 , ds .. .. . . dxn (s, r) = gn (t, x), xn (0, r) = rn . ds Solving these ordinary differential equations (o.d.e.’s), we obtain a parametrization of the variables t = t(s, r) and x = x(s, r). Now, using the equation that relates variables t and f in the chain of equations given in (15), one obtains the following o.d.e.: ∂f (s, r) = −f (t, x)∇x · g(t, x), f (0, r) = f0 (r), ∂s which has a well-known solution,   s  f (s, r) = f0 (r) exp − ∇x · g(σ, r) dσ .

(16)

0

To express the solution in terms of the original variables, we use the fact that our i.c.’s verify the hypotheses of Theorem 3. This result guarantees that we can invert the parametrization, hence obtaining the functions s(t, x) and r(t, x) that we can substitute, respectively, into expression (16). Therefore, the solution can be written as   t  ∇x · g(s(τ , x), r(τ , x)) dτ . (17) f (t, x) = f0 (r(t, x)) exp − t0

It is interesting to note that the function r(t, x) is actually the function we would obtain when solving the i.v.p. (4), say X(t) = h(t, X0 ), and then solving for X0 . With the corresponding notation, this writes as r(t, x) = h−1 (t, x) = x0 . As a consequence, the 1-p.d.f. given in (17) can be finally expressed in terms of the data    t    f (t, x) = f0 (x0 ) exp − ∇x · g(τ , x = h(τ , x0 )) dτ  t0

x0 =h−1 (t,x)

.

(18)

So far we have dealt with the case where randomness appears through i.c.’s only. Observe that the expression (18) is given in terms of the p.d.f., f0 , of the random i.c. X0 . Now, we extend this result to the general case where the r.d.e. also depends on a finite number of r.v.’s, represented by the absolutely continuous random vector A = (A1 , . . . , Am ), i.e.

Analysing Differential Equations with Uncertainties …

˙ X(t) = g(t, X(t), A), t > t0 , X(t0 ) = X0 ,

9

(19)

where g : [t0 , ∞[×L2n () × L2m () → L2n (). These kinds of random i.v.p.’s involving a finite number of r.v.’s (in this case n + m r.v.’s) are usually called i.v.p.’s with a finite degree of randomness [1, Chap. 3]. Henceforth, the p.d.f., f0 (x0 , a), of the random vector made up of all random inputs (X0 , A) is assumed to be known. Below, we shall show how to obtain a similar expression to (18) to the random i.v.p. (19), i.e. when uncertainties appear in i.c. and in the differential equation. To achieve this goal, the strategy will consist of transforming the random i.v.p. (19) into another random i.v.p. (having higher dimension), where randomness only appears through the i.c., and then we will apply (18). To this end, let us consider the extended random i.v.p. ˙ Y(t) = G(t, Y(t)), (20) Y(t0 ) = Y0 , where Y(t) = (X(t), A) = (X1 (t), . . . , Xn (t), A1 , . . . , Am ), Y0 = (X0 , A) and G(t, Y(t)) = (g(t, X(t)), 0) ∈ L2n+m () is continuously differentiable. With this new reformulation of i.v.p. (19), randomness only appears via the i.c. Y0 , and then the results shown in the first part of this section are applicable. Therefore, according to (12), the Liouville-Gibbs p.d.e. to the random i.v.p. (20) is given by ∂f (t, y) ∂ + (G k (t, y)f (t, y)) = 0. ∂t ∂yk n+m

(21)

k=1

Notice that G k (t, y) = gk (t, y), 1 ≤ k ≤ n, and G k (t, y) = 0, n + 1 ≤ k ≤ n + m. Therefore, taking into account that y = (x, a) ∈ Rn+m , p.d.e. (21) can be written as ⎧ n

∂ ⎪ ⎨ ∂f (t, x; a) + (gk (t, x, a)f (t, x; a)) = 0, t > t0 , ∂t ∂xk k=1 ⎪ ⎩ f (t0 , x; a) = f0 (x0 , a), where we have included the i.c., f0 (x0 , a), corresponding to the p.d.f. of the random initial vector (X0 , A). Then, the solution of this i.v.p. turns straightforwardly out   t   f (t, x; a) = f0 (x0 , a) exp − ∇x · g(τ , x = h(τ , x0 , a), a) dτ  t0

x0 =h−1 (t,x,a)

,

(22) where h(t, X0 , A) = X(t). Finally, the 1-p.d.f. of the solution s.p. to random i.v.p. (19) is derived by marginalising f (t, x; a) with respect to random vector A,  f (t, x) =

Rm

f (t, x; a) da.

(23)

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To conclude this section, we want to emphasise that the Liouville-Gibbs method is exact in the sense of providing a closed-form expression for the p.d.f. of the solution stochastic process in term of an integral (see expressions (18) and (22)– (23)) that can be exactly computable in some cases (otherwise accurate quadrature rules can be applied). An alternative exact method that can also be used is the socalled Random Variable Transformation (RVT) technique [1, Chap. 3]; however, its application requires defining an ad-hoc injective mapping having non-zero Jacobian that could become difficult to find in some situations. This fact may limit the use of RVT against the Liouville-Gibbs approach in some models. On the other hand, there exist alternate approaches like Monte Carlo simulations, but they only provide approximations in spite of eventually an exact solution is available. Furthermore, using the Monte Carlo method usually requires carrying out many simulations to obtain good approximations since its rate of convergence is slow [18] and this may become prohibitively demanding. All these comments will be pointed out in the forthcoming examples to inform the reader about the advantages of Liouville-Gibbs method against alternate approaches in the models that will be presented later.

3 Applications This section is addressed to showcase some relevant models, formulated via differential equations, that appear in different scientific realms. As in practice model parameters are usually fixed using sampling and/or measurements, so containing uncertainties, we will assume that they are r.v.’s instead of deterministic constants. This approach leads to formulate such models by means of r.d.e.’s. Then, we will take advantage of the main results exhibited in Sect. 2 to determine an explicit expression to the 1-p.d.f. of the solution s.p. of each randomised model, and afterwards we will compute the mean and the variance of the solution by integrating the corresponding 1-p.d.f. using expression (1). To show a full overview of the different scenarios treated in Sect. 2, firstly we will deal with the case that randomness only appears via the i.c. (see Sects. 3.1 and 3.2) where formula (18) applies. Secondly, we shall consider the general scenario where uncertainties appear in both i.c. and r.d.e. (see Sect. 3.3). In this case, we will take advantage of expressions (22)–(23).

3.1 The Random Linear Oscillator Certain physical processes, such as the dynamics of a mass-spring system or an electric current in a LC (Inductor-Capacitor) electronic circuit, can be modelled by a second-order o.d.e. with i.c.’s. In the mass-spring system, i.c.’s represent the initial position and velocity of the mass. In most physical experiments, these values are

Analysing Differential Equations with Uncertainties …

11

measured directly in the system and often measurement errors appear. Therefore, it is more natural to treat the i.c.’s as random variables rather than deterministic constants. Analogously, the initial electric current and voltage in an LC circuit model can be treated as r.v.’s. Let us consider the i.v.p. modelling the dynamics of an undamped linear oscillator X¨ (t) + ω 2 X (t) = 0, X (0) = X0 , X˙ (0) = X˙ 0 ,

(24)

where X0 , X˙ 0 are r.v.’s. We will assume that the joint p.d.f. of the i.c.’s, f0 (x0 , x˙ 0 ), is known. The (deterministic) parameter ω 2 > 0 represents the frequency of the oscillator. To take advantage of the results shown in Sect. 2, we introduce the change of variable X(t) = (X1 (t), X2 (t))T = (X (t), X˙ (t))T to transform model (24) into the following first-order random i.v.p.:   ⎧ ⎪ 0 1 ⎨X(t) ˙ = AX(t), A = , −ω 2 0 ⎪ ⎩ X(0) = X0 . Observe that according to (4), g(t, X(t)) = AX(t), g = (g1 , g2 ), being g1 (t, x) = x2 and g2 (t, x) = −ω 2 x1 , x = (x1 , x2 ), and X0 := (X0 , X˙ 0 )T . Therefore, ∇x · g(t, x) =

∂g1 (t, x) ∂g2 (t, x) + = 0, ∀(t, x) ∈]0, ∞[×R2 . ∂x1 ∂x2

As a consequence, the corresponding Liouville-Gibbs p.d.e. becomes ⎧ ⎨ ∂f (t, x) + x ∂f (t, x) − ω 2 x ∂f (t, x) = 0, 2 1 ∂t ∂y1 ∂x2 ⎩ f (0, x) = f0 (x0 , x˙ 0 ).

(25)

To compute f (t, x) = f (t, x, x˙ ), we first solve the random i.v.p. obtaining X(t) = eAt X0 , and then we solve for the random i.c., X0 = e−At X(t) =

   X0 cos(ωt) − ω1 sin(ωt) X (t) = . ω sin(ωt) cos(ωt) X˙ (t) X˙ 0

Therefore, according to (18) the solution of Liouville-Gibbs i.v.p. (24) writes f (t, x, x˙ ) = f0 (x0 , x˙ 0 )e−

t 0

0 ds

! = f0 x cos(ωt) −

x˙ ω

" sin(ωt), ωx sin(ωt) + x˙ cos(ωt) .

(26)

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V. Bevia et al.

Notice that by integrating this later expression, we can compute the 1-p.d.f. for the position and the velocity  f (t, x) =

+∞

−∞

 f (t, x, x˙ ) d˙x, f (t, x˙ ) =

+∞ −∞

f (t, x, x˙ ) dx,

(27)

respectively. In order to perform numerical simulations, we have chosen ω = 2 and the following Gaussian distributions for the i.c.’s, X0 ∼ N(0; 1) and X˙ 0 ∼ N(−1; 1). Assuming that they are independent, and if fX0 (x0 ) and fX˙ 0 (˙x0 ) denote their respective p.d.f.’s, we know that 1 − 1 (x02 +(˙x0 +1)2 ) . f0 (x0 , x˙ 0 ) = fX0 (x0 )fX˙ 0 (˙x0 ) = e 2 2π Then, according to (26) ! f (t, x, x˙ ) = f0 x cos(ωt) − =

x˙ ω

" sin(ωt), ωx sin(ωt) + x˙ cos(ωt)

 2 x˙ sin(tω) 1 − 21 (xω sin(tω)+˙x cos(tω)+1)2 1 − 2 x cos(tω)− ω e . 2π

(28)

By applying (27), we obtain the 1-p.d.f. of position,   (aω+sin(tω))2 exp − ω2 −1 ( ) cos(2tω)+ω2 +1 f (t, x) = √ # . 2 2π sinω(tω) + cos2 (tω) 2

(29)

Figure 1 shows its graphical representation. We can observe that the peaks in the 1-p.d.f. correspond to maximum amplitude in the oscillations. In the right panel of this plot, we show the approximations of the 1-p.d.f. obtained after applying Monte Carlo simulations with 10,000 simulations. The superiority of the Liouville-Gibbs method is clear because it provides more accurate results. Although theoretically similar results may be obtained using Monte Carlo at expense of increasing the number of simulations, this approach is demanding and the computational burden becomes unaffordable in comparison with the use of the Liouville-Gibbs approach. At this point, it is convenient to point out that this same model has been studied in [1, Chap. 6 (Example 6.1)], but using the r.v.t. method. However, the advantage of using the Liouville-Gibbs technique relies upon the fact that we do not need to look for an appropriate injective transformation whose Jacobian is distinct from zero to calculate the 1-p.d.f. Figure 2 shows the joint 1-p.d.f. of position and velocity in the phase space for fixed values of time, t = 0 and t = 1.3. They have been computed by expression (28). The red curve shows the orbit corresponding to the mean of the solution. The black dot corresponds, in each case, with the point of the orbit at t = 0 and t = 1.3.

Analysing Differential Equations with Uncertainties …

13

Fig. 1 Both plots correspond to the random i.v.p. (24) taking ω = 2, which implies an oscillation period of T = π units of time. Left panel: 1-p.d.f., f (t, x) given by (29), obtained by applying (27) to expression (26). Observe that the maxima of p.d.f.’s are reached at the points ( π2 k, ± 21 ), k = 0, 1, 2, . . ., which correspond when the oscillator reaches the maximum amplitude in its oscillations. Right panel: Approximation to the 1-p.d.f. given by a histogram (black bars) obtained by Monte Carlo sampling in the i.c.’s of i.v.p. (24). We have taken 10,000 simulations using the built-in random sampling tools by software Mathematica® . The solution in the right panel is overlayed to make the comparison easier. The superiority of Liouville-Gibbs approach is apparent against Monte Carlo simulations. Example of the random linear oscillator developed in Sect. 3.1

Notice that the mean in the phase diagram (position, X (t), and velocity, X˙ (t)) can be computed via  E[X (t)] =

+∞ −∞

xf (t, x) dx, E[X˙ (t)] =

where f (t, x) and f (t, x˙ ) are given by (27), or directly,



+∞

−∞

xf (t, x˙ ) d˙x,

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V. Bevia et al.

Fig. 2 Left: Joint p.d.f. of position and velocity at t = 0. Right: Joint p.d.f. of position and velocity at t = 1.3. As we can see, the fact that the level curves of the random i.v.p. (24) are ellipses, is in full agreement with properties of a binormal distribution. In both panels, the red curve represents the mean curve of the solution which is given by (30). Example of the random linear oscillator developed in Sect. 3.1

⎡ E[X(t)] = eAt E[X0 ] = ⎣ ⎡ =⎣

−itω

− ie2ω +

1 −itω e 2 1 itω ie ω 2

ieitω 2ω

− 21 e−itω − 21 eitω

⎤

ie−itω 2ω

+ 21 eitω

−

− 21 ie−itω ω 21 e−itω + 21 e 

⎤

 ⎦ E[X0 ] E[X˙ 0 ] itω

ieitω 2ω



(30)

⎦ = E[X (t)] , E[X˙ (t)]

where we have substituted E[X0 ] = 0 and E[X˙ 0 ] = −1. Finally, observe that in this example the solution s.p. can be expressed as X(t) = μX0 ; X0 ) being μ X0 = [0, M(t)X0 , where M(t) = eAt and X0 = (X0 , X˙ 0 )T ∼ N(μ −1]T and X0 = I2 (identity matrix of size 2, since X0 and X˙ 0 are independent Gaussian r.v.’s having each of them unit variance). Then, according to the wellknown properties of a linear transformation of a Gaussian random vector, the solution s.p. is also Gaussian (specifically it corresponds to a binormal distribution), μX(t) ; X(t) ), where X(t) ∼ N(μ   −itω itω 0 0 − ie2ω + ie2ω At = M(t) = =e −1 −1 − 21 e−itω − 21 eitω 

μ X(t) and

 X(t) = M(t)I2 M(t) = T

ω 2 +(ω 2 −1) cos(2tω)+1 2ω 2 ω 2 −1) sin(2tω) 1 −( 2ω 2

 ω 2 −1) sin(2tω) −( 2ω ! 2 ! 2 " " . ω − ω − 1 cos(2tω) + 1

Analysing Differential Equations with Uncertainties …

15

As a consequence, although initially both position and velocity are independent they become statistically dependent as time goes on. These features are also observed in the shape of the phase diagram depicted in Fig. 2.

3.2 The Random Damped Linear Oscillator In this section, we will extend the analysis performed in the previous example to the study the damped linear oscillator with random i.c.’s (position and velocity). It is known that its dynamics can be characterised by a homogeneous second-order linear differential equation, however for the sake of generality, we will first derive the 1-p.d.f. of the solution s.p. of nth order homogeneous differential equations of the form ⎧ n

⎪ ⎨ aj (t)X (j) (t) = 0, (31) j=0 ⎪ ⎩ (n−1) X (0) = X0 , . . . , X (0) = Xn−1 , dj X where X (j) (t) = j (t). We assume an (t) = 0, ∀t ≥ 0, and X0 , . . . , Xn−1 are r.v.’s. dt Also, we assume that the joint p.d.f. of the initial random vector X0 = (X0 , X1 , . . . , Xn−1 )T , denoted by f0 , is known. In order to take advantage of the results obtained in Sect. 2, we write (31) as n-dimensional first-order matrix o.d.e. To this end, we first divide by an (t) in (31) n−1

aj (t) (j) X (n) (t) = − X (t), a (t) j=0 n secondly, we put X(t) = (X (t), X  (t), . . . , X (n−1) (t))T , and then we obtain ⎡ ⎢ ⎢ ˙ X(t) = A(t)X(t), A(t) := ⎢ ⎣

0 0 .. .

1 0 .. .

0 1 .. .

··· ··· .. .

0 0 .. .

(t) (t) (t) − aa0n (t) − aa1n (t) − aa2n (t) · · · − aan−1 (t) n (t)

⎤ ⎥ ⎥ ⎥ , X(0) = X0 . ⎦

(32) It corresponds to the random i.v.p. (4) by taking t0 = 0, g(t, X(t)) = A(t)X(t). For t > 0 arbitrary but fixed, write X(t) = x = (x1 , . . . , xn ), then X (j) (t) = xj+1 , j ∈ {0, . . . , n − 1} and gj (t, x) = xj+1 , gn (t, x) = −

n−1

aj (t) xj+1 . a (t) j=0 n

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V. Bevia et al.

Therefore, ∂gj (t, x) = 0, ∀j ∈ {0, . . . , n − 1}, ∂xj and, as a consequence, ∇x · g(t, x) = −

∂gn (t, x) an−1 (t) , =− ∂xn an (t)

an−1 (t) . an (t)

Now we apply (18) to give an explicit expression of the 1-p.d.f. of the solution s.p. to the random i.v.p. (31) −1



f (t, x) = f0 (x0 = h (t, x)) exp 0

t

 an−1 (s) ds , an (s)

(33)

t

where h−1 (t, x) = e− 0 A(s) ds x, where A(s) is defined in (32). As an application of the previous result for the n-order linear o.d.e. with randomness in the i.c.’s, we will study the damped linear oscillator assuming that both its initial position and velocity are r.v.’s. Specifically, we will consider the following random i.v.p.: mX¨ (t) + cX˙ (t) + kX (t) = 0, t > 0, (34) X (0) = X0 , X˙ (0) = X˙ 0 , where m is the mass of the oscillator, c is called the viscous damping coefficient and k is a constant that depends on the oscillator [19, Sect. 6.1]. We will denote by f0 the joint p.d.f. of the initial random vector (X0 , X˙ 0 ). Using the same classical reasoning exhibited in the analysis of the damped linear oscillator, the i.v.p. (34) can be written in the form (4) where g(t, X(t)) = AX(t) being 

 0 1 c , k A= − − m m

(35)

X(t) = (X1 (t), X2 (t))T = (X (t), X˙ (t))T . As a consequence, g = (g1 , g2 ), with g1 (t, x) = x2 and g2 (t, x) = − mk x1 − mc x2 , x = (x1 , x2 ), and X0 := (X0 , X˙ 0 )T . Therefore, now ∇x · g(t, x) =

∂g1 (t, x) ∂g2 (t, x) c + =− , ∂x1 ∂x2 m

∀(t, x) ∈ [0, ∞[×R2 ,

and then the Liouville-Gibbs p.d.e. can be written as   ⎧ ⎨ ∂f (t, x) + x ∂f (t, x) − k x + c x ∂f (t, x) = c f (t, x), 2 1 2 ∂t ∂x1 m m ∂x2 m ⎩ f (0, x) = f0 (x0 , x˙ 0 ).

Analysing Differential Equations with Uncertainties …

17

It is well known that the solution of this model, X(t) = eAt X0 , can be expressed √ √ c2 −4km c2 −4km , λ2 = −c− 2m in different forms depending on the eigenvalues λ1 = −c+ 2m of matrix A defined in (35). It leads to three different physical behaviours of the oscillator, namely, 1. If c2 > 4km, the oscillator will be overdamped. In such case, 0 > λ1 > λ2 . The system returns to its steady state (equilibrium state) without oscillating. 2. If c2 = 4km, the oscillator will be critically damped. In such case, 0 > λ1 = c . The system returns to its equilibrium state as quickly as possible λ2 = − 2m without oscillating, although overshoot (1 oscillation) may occur. 3. If c2 < 4km, the oscillator will be underdamped. In such a case, we will have two conjugate complex roots. The system oscillates, but the amplitude gradually decays to 0. For illustrative purposes only, hereinafter, we will consider the third case which is usually referred to as the damped linear oscillator. In such a case, the solution is given by Y(t) = eAt Y0 and then Y0 = e−At Y(t) ⎛

⎡

⎢ e ⎝cos ⎢ =⎢ ⎢ ⎣ ct m

√

4km−c2 t m



−

√ ⎞ 4km−c2 t c sin m ⎠ √ 4km−c2

 √ ct 4km−c2 t e m k sin m √ 4km−c2

√  ct 4km−c2 t e m m sin m √ − 4km−c2

√

ct

e m cos

4km−c2 t m

⎤

⎥ ⎥ ⎥ Y(t). ⎥ ⎦

Therefore, using (33), and identifying an (t) = m, an−1 (t) = c and an−2 (t) = k in (31) (with n = 2), we obtain the 1-p.d.f. of the solution s.p. after re-substituting (y1 , y2 ) = (x, x˙ ) ⎛

⎛

f0 ⎝xe ⎝cos ct m

x



t

√

4km−c2 m

  √ 2 ct ke m sin t 4km−c m √ 4km−c2



−

⎞  √ 2 c sin t 4km−c m ⎠ √ 4km−c2

ct m

+ x˙ e cos



t

√

4km−c2 m

−

  √ 2 ct me m sin t 4km−c m √ , x˙ 4km−c2

⎞  ⎠ e ctm = f (t, x, x˙ ).

(36)

To carry out numerical simulations, we will assume that the i.c.’s are independent Gaussian r.v.’s, X0 ∼ N(0; 0.0625), X˙ 0 ∼ N(−1; 0.25). In Fig. 3, we show the marginal 1-p.d.f. of position, f (t, x), on the time interval t ∈ [0, 10], using  f (t, x) = where f (t, x, x˙ ) is given by (36).

+∞

−∞

f (t, x, x˙ ) d˙x,

(37)

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V. Bevia et al.

Fig. 3 Level curves of the 1-p.d.f. of position, f (t, x) for t ∈ [0, 10]. For each t, we see that the p.d.f. takes its highest values when the oscillator reaches its maximum amplitude and that the amplitude gradually decreases. This is a different, but equivalent representation to Fig. 1 in our new setting oscillator model. Example of the random damped linear oscillator in Sect. 3.2. Function f (t, x) has been computed using the ‘Global Adaptive’ built-in integration method by software Mathematica®

3.3 The Random Logistic Model In this subsection, we study the randomised version of the logistic differential equation, which plays a very important role in modelling systems whose behaviour is characterised by a first stage with rapid growth followed by a slower growth until reaching stabilisation. Particularly, this model is very useful in biology when describing the dynamics of a population whose individuals have limited available resources. The logistic model has also been used to describe the diffusion (demand) of a certain technology (mobile phones, use of electronic commerce, etc.) characterised by a fast initial growth and whose long-term growth is slow down due to the extensive use of the given technology by all the individuals of the population, saturating its demand. Here, we will consider a fully randomised formulation of the logistic model via the following i.v.p.: X  (t) = X (t)(A − BX (t)), t > t0 , (38) X (t0 ) = X0 ,

Analysing Differential Equations with Uncertainties …

19

where X0 , A and B are assumed to be absolutely continuous r.v’s. Here, A denotes the random reproductive parameter of the population (also termed growth rate), B is the ratio between the growth rate A and the maximum, or asymptotic, population (also termed random carrying capacity) and X0 the initial number of individuals. For the sake of generality, in our subsequent theoretical analysis we assume that the initial input random vector (X0 , A, B) has a joint p.d.f., f0 (x0 , a, b), which is assumed to be known. According to the notation introduced for the extended i.v.p. (20), in this case n = 1 and m = 2, being Y(t) = (X (t), A) = (X (t), A, B), G(t, Y(t)) = (g(t, X (t)), 0, 0) and X0 = (X0 , A) = (X0 , A, B). Besides, observe that g(t, x) = x(a − bx), thus ∇x · g(t, x, a, b) = a − 2bx. Therefore, the Liouville-Gibbs p.d.e. (21) becomes ⎧ ∂f (t, x) ⎨ ∂f (t, x) + x(a − bx) = −(a − 2bx)f (t, x), t > t0 , ∂t ∂x ⎩f (t , x) = f (x , a, b). 0

0

0

On the other hand, it is well known that the solution of random i.v.p. (38) is given by X (t) =

X0 +

A X B 0 e−A(t−t0 ) ( BA

− X0 )

= h(t, X0 , A).

(39)

Let us fix t > t0 and denote X (t) = X . Solving (39) for X0 , that is, obtaining the inverse of h as a function of X0 gives X0 =

X+

A X B A(t−t 0)( A e B

− X)

= h−1 (t, X , A).

In order to apply expression (22), we first need to calculate the following integral: 

t

 ∇x · g(s, h(t, x0 , a)) ds =

t

∇x · g(s, h(t, x0 , a, b)) ds   a2 = −a(t − t0 ) + ln . (bx0 + ea(t−t0 ) (a − bx0 ))2

t0

t0

Now, using (22), we obtain 

 , a, b

a2 ea(t−t0 ) . x+ − x) (bx + ea(t−t0 ) (a − bx))2 (40) Finally, we marginalise with respect to A = (A, B) using (23), and we then obtain the 1-p.d.f. of the solution s.p. of random i.v.p. (38) a x b ea(t−t0 ) ( ab

f (t, x; a) = f (t, x; a, b) = f0

 f (t, x) =

+∞ −∞



+∞ −∞

f (t, x; a, b) da db.

(41)

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To illustrate our previous theoretical conclusions, we will assume the following distributions for the random inputs: X0 ∼ N|[0,1] (0.05, 0.0001) (a Gaussian distribution truncated on the interval [0, 1]), and uniform distributions for A and B, specifically A ∼ Un(0.3, 0.5) and B ∼ Un(2, 3). Assuming that X0 , A and B are independent, the factor f0 in (40) can be expressed as  f0

x+

a x b ea(t−t0 ) ( ab

− x)

  , a, b = fX0

x+

a x b ea(t−t0 ) ( ab

 − x)

fA (a)fB (b),

(42)

being fX0 , fA and fB the p.d.f.’s of X0 , A and B, respectively. In Fig. 4, we show the 1-p.d.f. of the solution X (t) for different values of t. This graphic has been performed using expressions (40)–(42). We can observe as the value of the mean and the variance increase until stabilisation. This behaviour is in full agreement with the graphical representation shown in Fig. 5. Table 1 collects the values of the mean and the variance of the solution s.p., E[X (t)], at the same instants t = 0, 1, 5, 10, 20, 40 shown in Fig. 4. These figures have been calculated by expression (1) (with k = 1) and applying (40)–(42). Observe that these numerical values agree with the ones observed in Figs. 4 and 5. As we can see in Table 1, the limit of the mean of the solution s.p. converges to 0.1624 approximately. This fact can be rigorously checked in a different way. From (39), observe that X (t) −−−→ A/B, and applying the r.v.t. t→∞ method [1, Chap. 2] it can be seen that the p.d.f. of the limit r.v., Z := A/B, is given by  fZ (z) =

+∞

−∞

fA (ξz)fB (ξ) dξ, 

then E[A/B] = E[Z] =

+∞

−∞

zfZ (z) dz.

One of the most useful applications of determining an expression of the 1-p.d.f., f (t, x), is the computation of confidence intervals for a specific confidence level (1 − α) × 100% for each fixed time ˆt . These intervals are computed by determining a suitable value kˆt > 0 such that (

)

1 − α = P {ω ∈  : X (ˆt , ω) ∈ [L(ˆt ), U (ˆt )]} =



U (ˆt )

L(ˆt )

f (ˆt , x) dx,

(43)

being L(ˆt* ) := μX (ˆt ) − kˆt σX (ˆt ), U (ˆt ) := μX (ˆt ) + kˆt σX (ˆt ), and μX (ˆt ) = E[X (ˆt )] and σX (ˆt ) = V[X (ˆt )] the mean and the standard deviation, respectively. In our computations, we have taken α = 0.05 to construct 95% confidence intervals. Results are shown in Fig. 5. In this plot, we can observe that the diameters of the confidence intervals tend to stabilise as t → ∞. This feature is in full agreement with the stabilisation of variance previously shown.

Analysing Differential Equations with Uncertainties …

21

Fig. 4 Time evolution of the 1-p.d.f. of the solution s.p. of random logistic equation (38) for different time instants. It can be seen how variance grows until it stabilises. In fact, the difference between the p.d.f. corresponding to t = 20 and t = 40 is barely observable because both p.d.f.’s overlap having as mean value 0.1624. Example of the random logistic differential equation developed in Sect. 3.3. Function f (t, x), given by Eq. (41), has been computed using the ‘Multidimensional Rule’ built-in method by software Mathematica®

Fig. 5 Time evolution of the mean, μX (t), and the confidence interval, μX (t) ± kt σX (t), where kt > 0 and σX (t) is the standard deviation of the solution s.p. X (t). The coefficient kt > 0 has been determined, for each t > 0, so that the confidence interval captures 95% of probability according to (43). Example of the random logistic differential equation developed in Sect. 3.3. We have obtained the values kt by solving the non-linear equations that appear in (43) using the numerical root finder tool and the ‘Multidimensional Rule’ built-in integration method by software Mathematica®

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Table 1 Mean and variance of the solution s.p. of i.v.p. (38) at different time instants. We observe that the mean tends to 0.1624 whilst the variance stabilises around 9.13 × 10−4 . Example of the random logistic differential equation developed in Sect. 3.3 t=0 t=2 t=5 t = 10 t = 20 t = 40 t = 100 E[X (t)] V[X (t)]

0.05 9.93 × 10−5

0.0802289 0.123783 2.05 × 5.76 × 10−4 10−4

0.155381 9.15 × 10−4

0.162014 9.17 × 10−4

0.162117 9.13 × 10−4

0.162383 9.13 × 10−4

4 Conclusions In this contribution, we have shown the key role played by the Liouville-Gibbs equation to determine the first probability density function of the solution stochastic process of a first-order random matrix differential equation whose uncertainties may appear in both the initial condition and its coefficients via a finite number of random variables (finite degree of randomness). This approach has important advantages with respect to the so-called Random Variable Transformation method which is often applied to get the aforementioned goal. Indeed, unlike the Random Variable Transformation method, the Liouville-Gibbs approach avoids defining ad-hoc mappings and calculating the Jacobian of the transformation, and therefore reducing the computational cost involved when obtaining the probability density function. The Liouville-Gibbs theorem also explicitly provides a partial differential equation satisfied by the probability density function of the solution stochastic process. Our contribution has been completed by showing the application of the Liouville-Gibbs equation to several problems belonging to Physics and Biology that illustrate its usefulness in dealing with models formulated via random differential equations with a finite degree of uncertainties. Acknowledgements This work has been supported by the Spanish Ministerio de Economía, Industria y Competitividad (MINECO), the Agencia Estatal de Investigación (AEI) and Fondo Europeo de Desarrollo Regional (FEDER UE) grant MTM2017-89664-P, and by the European Union through the Operational Program of the European Regional Development Fund (ERDF)/European Social Fund (ESF) of the Valencian Community 2014–2020, grants GJIDI/2018/A/009 and GJIDI/2018/A/ 010. Ana Navarro Quiles acknowledges the funding received from Generalitat Valenciana through a postdoctoral contract (APOSTD/2019/128). Computations have been carried thanks to the collaboration of Raúl San Julián Garcés and Elena López Navarro granted by European Union through the Operational Program of the European Regional Development Fund (ERDF)/European Social Fund (ESF) of the Valencian Community 2014–2020, grants GJIDI/2018/A/009 and GJIDI/2018/A/010, respectively.

Analysing Differential Equations with Uncertainties …

23

References 1. Soong, T.T.: Random Differential Equations in Science and Engineering. Academic Press, New York (1973) 2. Henderson, T.T., Plaschko, P.: Stochastic Differential Equations in Science and Engineering. World Scientific, New Jersey (2006) 3. Neckel, T., Rupp, F.: Random Differential Equations in Scientific Computing. Walter de Gruyter, München, Germany (2013) 4. Cortés, J.-C., Navarro-Quiles, A., Romero, J.-V., Roselló, M.-D.: Randomizing the parameters of a Markov chain to model the stroke disease: a technical generalization of established computational methodologies towards improving real applications. J. Comput. Appl. Math. 324, 225–240 (2017). https://doi.org/10.1016/j.cam.2017.04.040 5. Cortés, J.-C., Navarro-Quiles, A., Romero, J.-V., Roselló, M.-D.: Full solution of random autonomous first-order linear systems of difference equations. Application to construct random phase portrait for planar systems. Appl. Math. Lett. 68, 150–156 (2017). https://doi.org/10. 1016/j.aml.2016.12.015 6. Dorini, F.A., Cecconello, M.S., Dorini, L.B.: On the logistic equation subject to uncertainties in the environmental carrying capacity and initial population density. Commun. Nonlinear Sci. Numer. Simul. 33, 160–173 (2016). https://doi.org/10.1016/j.cnsns.2015.09.009 7. Casabán, M.-C., Cortés, J.-C., Navarro-Quiles, A., Romero, J.-V., Roselló, M.-D., Villanueva, R.-J.: A comprehensive probabilistic solution of random SIS-type epidemiological models using the random variable transformation technique. Commun. Nonlinear Sci. Numer. Simul. 32, 199–210 (2016). https://doi.org/10.1016/j.cnsns.2015.08.009 8. Hussein, A., Selim, M.M.: Solution of the stochastic radiative transfer equation with Rayleigh scattering using RVT technique. Appl. Math. Comput. 218(13), 7193–7203 (2012). https://doi. org/10.1016/j.amc.2011.12.088 9. Hussein, A., Selim, M.M.: Solution of the stochastic transport equation of neutral particles with anisotropic scattering using RVT technique. Appl. Math. Comput. 213(1), 250–261 (2009). https://doi.org/10.1016/j.amc.2009.03.016 10. Loève, M.: Theory of Probability. Springer, New York (1973) 11. Villafuerte, L., Braumann, C., Cortés, J.-C., Jódar, L.: Random differential operational calculus: theory and applications. Comput. Math. Appl. 59(1), 115–125 (2010). https://doi.org/10.1016/ j.camwa.2009.08.061 12. Gasquet, C., Witomski, P.: Fourier Analysis and Applications: Filtering, Numerical Computation, Wavelets. Springer, New York (1991) 13. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (2001) 14. Koch, G., Nadirashvili, N., Seregin, G.A., ver(mohana)k, V.: Liouville theorems for the NavierStokes equations and applications. Acta Mathematica 203, 83–105 (2009). https://doi.org/10. 1007/s11511-009-0039-6 15. Folland, G.B.: Introduction to Partial Differential Equations. Princeton University Press, New Jersey (1995) 16. Delgado, M.: The Lagrange-Charpit method. SIAM Rev. 39(2), 298–304 (1997). https://doi. org/10.1137/S0036144595293534 17. Farlow, S.J.: Partial Differential Equations for Scientists and Engineers. Dover, New Jersey (1993) 18. Xiu, D.: Numerical Methods for Stochastic Computations: A Spectral Method Approach. Princeton University Press (2010) 19. Strang, G.: Introduction to Applied Mathematics. Wellesley-Cambridge Press (1986)

Solving Time-Space-Fractional Cauchy Problem with Constant Coefficients by Finite-Difference Method Reem Edwan, Rania Saadeh, Samir Hadid, Mohammed Al-Smadi, and Shaher Momani

Abstract In this chapter, we present the time-space-fractional Cauchy equation with constant coefficients, the space and time-fractional derivative are described in the Riemann-Liouville sense and Caputo sense, respectively. The implicit scheme is introduced to solve time-space-fractional Cauchy problem in a matrix form by utilising fractionally Grünwald formulas for discretization of Riemann-Liouville fractional integral, and L1-algorithm for the discretization of time-Caputo fractional derivative, additionally, we provided a proof of the von Neuman type stability analysis for the fractional Cauchy equation of fractional order. Several numerical examples are introduced to illustrate the behaviour of approximate solution for various values of fractional order.

1 Introduction Many phenomena in non-Brownian motion, fluid flows, chemical science, management theory, signal process, fibre optics, systems identification, elastic materials, polymers and others, are well described by a fractional differential equation. In specific, the partial differential equations (PDE) of fractional order are progressively R. Edwan (B) Department of Mathematics, Faculty of Science, Taibah University, Al-Madinah al-Munawarah, Medina, Saudi Arabia e-mail: [email protected] R. Saadeh Department of Mathematics, Faculty of Science, Zarqa University, Zarqa 13110, Jordan S. Hadid · S. Momani Department of Mathematics and Sciences, College of Humanities and Sciences, Ajman University, Ajman, UAE M. Al-Smadi Department of Applied Science, Ajloun College, Al-Balqa Applied University, Ajloun, Jordan © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 D. Zeidan et al. (eds.), Computational Mathematics and Applications, Forum for Interdisciplinary Mathematics, https://doi.org/10.1007/978-981-15-8498-5_2

25

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used to model issues in finance, viscoelasticity, mathematical biology and chemistry [1–8]. Different partial differential equations of fractional order are studied and resolved by several powerful methods [9–17]. Consequently, considerable attention has been given to the answer of partial differential equations of fractional order. Several powerful strategies are established and developed to induce numerical and analytical solutions of fractional differential equations, like finite-difference technique [7], finite volume technique [9], finite element technique [11], homotopy perturbation technique [13] and the fractional sub-equation technique [2]. Recently, many scholars introduced methods for solving fractional differential equations. Momani developed a domain decomposition technique to approximate solution for the fractional convection-diffusion equation with a nonlinear source term [12]. Dehghan et al introduced a numerical solution for a class of fractional convection-diffusion equations using the flatlet oblique multi-wavelets [8]. Saadatmandi et al studied the sinc-Legendre collocation technique for a category of fractional convection-diffusion equations with variable coefficients [14]. Liu et al introduced the finite volume technique for solving the fractional diffusion equations [18], and Yang et al proposed the finite volume technique to the fractional diffusion equations [19], all of that are without theoretical analysis. Meerschaert and Tadjeran proposed the finite-difference technique for the resolution of the fractional advection-dispersion flow equations [16]. Baeumer and Meerschaert obtained the solution for fractional Cauchy equations by subordinating the solution of the original Cauchy equation [20]. Pskhu introduced a fundamental solution of a higher order Cauchy equation with time-fractional derivative [21]. Recently, Hejazi et al utilised the finite volume technique and finite-difference technique for solving the space-fractional advection-dispersion equation [17]. They used fractionally shifted Grünwald formula for the fractional derivative and verified the stability and convergence of the scheme, whose order is O(τ + h). During this Chapter, we propose a finite-difference technique to get a new approximate solution for the time-space-fractional Cauchy equation with constant coefficients, space-fractional derivative and time-fractional derivative are described within the Riemann-Liouville sense and Caputo sense, respectively. Consider the time-space-fractional Cauchy equation of the shape ∂ β u(x, t) ∂ γ u(x, t) + = g(x, t), γ ∂t ∂xβ

(1)

subject to the initial condition u(x, 0) = f (x), where t > 0, x ∈ [a, b], 0 < β ≤ 1, 0 < γ ≤ 1, g(x, t) is a given function provided that u(x, t), g(x, t), and f (x) are smooth enough,  is a positive parameter,

Solving Time-Space-Fractional Cauchy Problem …

27

β is a parameter describing the order of the space fractional, and γ is a parameter describing the order of the time-fractional, the space-fractional derivative and timefractional derivative are described in the Riemann-Liouville sense and Caputo sense, respectively. The starting point for a finite-difference discretization is a partition of the computational domain [a, b] into a finite number of sub-domains Vi , i = 0, 1, 2, . . . , N , known as control volumes CVs, the union of all CVs should cover the whole domain. We introduce the implicit scheme by discretization of the Riemann-Liouville fractional integral, and time-Caputo fractional derivative. For another numerical scheme, see [22–31]. This chapter introduces a finite-difference technique for solving the time-spacefractional Cauchy equation with constant coefficients and contains the following sections: Sect. 2 is devoted to mathematical preliminaries. The description of a modified finite-difference technique is presented in Sect. 3. The von Neuman type stability analysis and consistency are proved in Sect. 4. Whilst the numerical experiments are given in Sect. 5. Finally, a brief conclusion is outlined in the last section.

2 Preliminaries Throughout the past decade, fractional calculus has been applied to virtually every field of engineering, economics, science and another field. People like Liouville, Riemann and Weyl created major contributions to the idea of fractional calculus [32– 39]. The story of the fractional calculus continued with contributions from Fourier, Abel, Leibniz, Grünwald and Letnikov. Over the years, several definitions found that are acceptable for the concept of fractional derivatives and integrals [40–48]. Definition 1 The Riemann-Liouville integral of fractional order α > 0, Jaα u(x) is defined by Jaα u(x) =

1 Γ (α)



x

(x − t)α−1 u(t)dt, t > a,

a

provided that u ∈ L 1 [a, b]. For α = 0, we have Ja0 u(x) = u(x) is the identity operator. Definition 2 Let n ∈ N be the smallest integer that exceeds α, then the RiemannLiouville fractional derivative of order α > 0 is defined by Daα u(x)

 n  x  d u(t) 1 = dt . α+1−n Γ (n − α) d x n a (x − t)

(2)

provided that Daα u(x) = Dn Ja(n−α) u(x). For α = 0, we have Da0 u(x) = u(x) is the α . identity operator. For α ∈ N, Daα u(x) = d du(x) xα

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Definition 3 Let α > 0, u ∈ C α [a, b]. Then,   1 [ x−a h ] α k α  Da u(x) = lim α u(x − kh), (−1) k=0 h→0 h k

(3)

is called the Grünwald-Letnikov fractional where a < x ≤ b, with h = x−a N derivative of order α of the function u. Definition 4 Let n be the smallest integer that exceeds α, then the Caputo fractional derivative of order α > 0 is defined by D∗α u(x)

=

1 Γ (n−α) n d u(x) , dxn

x

u (n) (t) 0 (x−t)α+1−n ,

n − 1 < α < n, α = n,

(4)

provided that D∗α u(x) = Da−(n−α) Dn u(x) whenever Dn u ∈ L 1 [a, b]. The following theorem shows the relation between this definition and the Riemann-Liouville fractional derivatives: Theorem 1 Let α > 0, n = α and u ∈ C n [a, b]. Then, aα u(x) = Daα u(x), a < x ≤ b. D Theorem 2 Let α > 0, and u ∈ C[a, b]. Then, J aα u(x) = lim h α

[ x−a h ] k=0

h→0

 (−1)k

 x −a −α , u(x − kh), h = k N

(5)



 −α Γ (α+k) where a < x ≤ b, (−1) = Γ (α)Γ , and the = α(α−1)(α−2)...(α−k+1) k! (k+1) k

∞ x−1 −t function (x) is defined by (x) = 0 t e dt . α If we define weights w0α = 1, w1α = α and wkα = 1 − (1−α) wk−1 , k = 2, 3, . . . , k then we may rewrite (5) as k

J aα u(x) = lim h α

[ x−a h ]

h→0

k=0

wkα u(x − kh),

This formula is used to approximations the fractional integrals Jaα u(x). Lemma 1 Let 0 < α < 1. Then, we have (1) w0α = 1, and wαj > 0 for j = 1, 2, . . .; (2) wαj − wαj+1 > 0 for j = 0, 1, . . .; (3) lim j→∞ wαj = 0.

(6)

Solving Time-Space-Fractional Cauchy Problem …

29

Proof For the first part, let w0α = 1 and w1α = α > 0, thus from the recursive definition   (1 − α) α w j−1 , k = 2, 3, . . . , (7) wαj = 1 − j and since 0 < α < 1, we have 0 < 1−α < 1j < 1 for j ≥ 2. So the coefficient j  1 − (1−α) in (7) is strictly between zero and one. k Now, the second part can be done for j ≥ 2 such that     (1 − α) α (1 − α) α w j−1 − 1 − wj wαj − wαj+1 = 1 − j j +1      (1 − α) α (1 − α) α (1 − α) w j−1 − 1 − 1− w j−1 = 1− j j +1 j     (1 − α) α (1 − α) 1− w j−1 = 1− 1− j +1 j   (1 − α) α (1 − α) 1− w j−1 > 0. = j +1 j Finally, from 1 and 2 we have for j ≥ 2 0 < wαj+1 < wαj < w1α = α < 1 = w0α . So, lim wαj = 0. j→∞

Whenever we use a numerical technique to solve a differential equation, we would like to make sure that the numerical solution obtained is a sufficiently good approximation to the actuality solution, some necessary definition and remarks are introduced to discuss the stability analysis [27, 49–54]. To analyse the stability of difference scheme for IVP, suppose that we are given a vector in 2 , v = (. . . , v−1 , v0 , v1 , . . .)T , and define the discrete fourier transform of v as follows:

Definition 5 The discrete Fourier transform of v ∈ 2 is the operation v ∈ L 2 [−π, π ] defined by ∞  1 v (ξ ) = √ e−iξ vm , ξ ∈ [−π, π ]. 2π m=−∞



n

Definition 6 The symbol of difference scheme vn+1 = Qvn is the coefficient of v in n+1 n n+1 n = ρ(ξ )v , where v = ρ(ξ )v is the discrete Fourier transform the equation v of the discrete scheme.





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Remark 1 For simplification, we can get the discrete Fourier transform of the difference scheme by replacing vnj in the difference scheme by n

vnj = v ex p(i jξ ), i =

√

−1.

Remark 2 The difference scheme vn+1 = Qvn is stable with respect to 2,h norm if and only if there exist positive constants τ0 , h 0 and  so that |ρ(ξ )| ≤ 1 + τ , for 0 < τ ≤ τ0 , 0 < h ≤ h 0 and all ξ ∈ [−π, π ]. Remark 3 If ρ satisfies the inequality in Remark 2, then ρ is said to be satisfied the von Neumann condition. Remark 4 The difference scheme that is stable under a set of conditions is called conditionally stable, otherwise is called unconditionally stable scheme.

3 Modified Finite-Difference Method In this section, we propose a new finite-difference method for solving the time-spacefractional Cauchy equation of the shape: ∂ β u(x, t) ∂ γ u(x, t) + = g(x, t), γ ∂t ∂xβ

(8)

subject to the initial condition u(x, 0) = f (x) for t > 0, x ∈ [a, b], 0 < β ≤ 1, 0 < γ ≤ 1, g(x, t) is a given function provided that u(x, t), g(x, t), and f (x) are smooth enough,  is a positive parameter, β is a parameter describing the order of the space fractional and γ is a parameter describing the order of the time-fractional, the space-fractional derivative and time-fractional derivative are described in the Riemann-Liouville sense and Caputo sense, respectively. Using the definition of Riemann-Liouville fractional derivative where 0 < β ≤ 1, we have ∂ ∂ γ u(x, t) +  Ja1−β u(x, t) = g(x, t), ∂t γ ∂x 1−β

(9)

where Ja is the Riemann-Liouville integral with respect to x, take α = 1 − β, we have 0 ≤ α < 1. Let  = [a, b] be a finite domain that is discretised with N + 1 uniformly spaced nodes xi = a + i h, i = 0, 1, . . . , N , where the spatial , we approximate the α order fractional Riemann-Liouville integral step h = b−a N with standard Grünwald formula and approximate the first derivative with central difference formula:

Solving Time-Space-Fractional Cauchy Problem …

Jaα u(x, t) = h α

N 

wαj u(x − j h, t) + o(h),

31

(10)

j=0

   ∂u(x, t)  u(xi+1 , t) − u(xi−1 , t) + O h2 . =  ∂x 2h x=xi

(11)

A finite-difference discretization is applied by evaluating Eq. (9) at x = xi , and using the above equations. ⎤ ⎡ i+1 i−1     ∂ γ u(xi , t)  ⎣ α α  =− w j u xi− j+1 , t − h α wαj u xi− j−1 , t ⎦ + g(xi , t). h ∂t γ 2h j=0 j=0 Let tn = nτ , n = 0, 1, 2, . . ., where τ is the time step, and discretise the Caputo time-fractional derivative using L1-algorithm,      τ −γ ∂ γ u(xi , tn+1 ) = bγ u(xi , tn+1−s ) − u(xi , tn−s ) + O τ 2−γ . (12) ∂t γ Γ (2 − γ ) s=0 s n

γ

where bs = (s + 1)1−γ − s 1−γ , s = 0, 1, . . . , n. Letting u in ≈ u(xi , tn ) denote the numerical solution, we have n    τ −γ bγ u n+1−s − u in−s Γ (2 − γ ) s=0 s i ⎤ ⎡ i+1 i−1   ⎣ α  α n+1 n+1 ⎦ w j u i− j+1 − h α wαj u i− =− h j−1 + g(x i , tn+1 ). 2h j=0 j=0

(13)

Collecting like terms, we can rewrite Eq. (13) as: n N    1 τ −γ bsγ u in+1−s − u in−s = bi j u n+1 + gin+1 , j Γ (2 − γ ) s=0 h j=0

(14)

 ⎧ α α α h wi− j+1 −wi− j−1 ⎪ ⎪ , j < i − 1, ⎪ ⎪ 2α ⎪ α α ⎪ ⎨ h [w2 −w0 ] , j = i − 1, 2 where i = 0, 1, . . . , N , and bi j = h α w1α , j = i, ⎪ 2 ⎪ ⎪ h α w0α ⎪ , j = i + 1, ⎪ ⎪ ⎩ 2 0, j > i + 1.  n n  n n = u , u Denoting the numerical solution vector U 0 1 , . . . , u N and source vector   g n+1 = g(x0 , tn+1 ), g(x1 , tn+1 ), . . . , g(x N , tn+1 ) , we have the following vector equation:

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 I+

 n−1   γ Γ (2 − γ )τ γ γ  bs − bs+1 U n−s + τ γ Γ (2 − γ )g n+1 , A U n+1 = bnγ U 0 + h s=0

where the matrix A has elements ai j = bi j . In particular, for γ = 1 we can use the standard backward difference to approximate the time derivative in Eq. (12)  u(xi , tn+1 ) − u(xi , tn ) du(xi , t)  = + O(τ ),  dt τ t=tn+1

(15)

yields the numerical solution − τ

u in+1

⎡ u in

=−

 ⎣ α h 2h

i+1 

n+1 α wαj u i− j+1 − h

j=0

⎤

i−1 

n+1 ⎦ n+1 wαj u i− , j−1 + gi

(16)

j=0

Anyhow, we can rewrite Eq. (16) as a vector equation in the form 

I+

τ n+1 = U n + τ g n+1 , A U h

(17)

where the matrix A has elements ai j = bi j . In the next section, we prove that this scheme is conditionally stable, and it is first-order accurate in time and second-order accurate in space.

4 Stability Analysis In this section, the stability analysis for the proposed numerical scheme is presented as in the following theorems: Theorem 3 The numerical scheme (16) is conditionally stable. Proof To debate stability, consider the homogeneous scheme N n u n+1 1 m − um bm j u n+1 = j , m = 0, 1, 2, . . . , N . τ h j=0 n

Substitution of u nm = u ex p(imξ ), i = n+1

u

√

ex p(imξ ) − u n ex p(imξ ) = r

−1 into numerical scheme

N  j=0

n+1

bm j u

exp(i jξ ), r =

τ . h

Solving Time-Space-Fractional Cauchy Problem … n+1

u

33

1  un . = N 1 − r j=0 bm j exp(i( j − m)ξ )

The symbol of numerical scheme is 1 , ρ(ξ ) =  N 1 − r j=0 bm j ex p(i( j − m)ξ ) and satisfies the von Neumann condition if     N    1 − r bm j ex p(i( j − m)ξ ) ≥ 1.    j=0 By using Reverse Triangle Inequality, we have got       N     N          1 − r  bm j ex p(i( j − m)ξ ) ≥ 1 − r bm j exp(i( j − m)ξ ).   j=0     j=0         N   So, the von Neumann condition satisfies if 1 − r bm j ex p(i( j − m)ξ ) ≥  j=0        N 1. That is equivalent to 1 − r j=0 bm j ex p(i( j − m)ξ ) ≥ 1, impos     sible hold, or 1 − r Nj=0 bm j ex p(i( j − m)ξ ) ≤ −1.which is equivalent to    N   j=0 bm j ex p(i( j − m)ξ ) ≥ r2 . Hence, the symbol of numerical scheme satis    fies the von Neumann condition if  Nj=0 bm j ex p(i( j − m)ξ ) ≥ r2 .∀m = 0, 1, 2, . . . , N . So, the numerical scheme is conditionally stable. Theorem 4 The numerical scheme (16) is consistent with second-order accuracy in direction of space and first order in direction of time. Proof By using Eqs. (10), (11) and (15), we can write Eq. (8) at (xi , tn+1 ) as follows: u in+1 − u in + O(τ ) τ ⎡

⎤ i+1 i−1   2  ⎣ α  α n+1 n+1 ⎦ =− + gin+1 . w j u i− j+1 + o(1) − h α wαj u i− h j−1 + o(1) + O h 2h j=0 j=0 Thus, we get that

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   ∂u(xi , t)  ∂  1−β  J = − u(x, t + gin+1 , ) n+1  a  ∂t ∂ x t=tn+1 x=xi which is Eq. (8) at (xi , tn+1 ), γ = 1. Theorem 5 The numerical scheme (14) is consistent with second-order accuracy in space and 2-γ order in time. Proof we can write Eq. (8) at (xi , tn+1 ) as follows: n      τ −γ bsγ u(xi , tn+1−s ) − u(xi , tn−s ) + O τ 2−γ Γ (2 − γ ) s=0 ⎤ ⎡ i+1 i−1   2  ⎣ α  α n+1 n+1 ⎦ + gin+1 w j u i− j+1 + o(1) − h α wαj u i− =− h j−1 + o(1) + O h 2h j=0 j=0

Thus, we have    ∂ γ u(xi , t)  ∂  1−β Ja u(x, tn+1 )  = − + gin+1 .  ∂t γ ∂ x t=tn+1 x=xi which is Eq. (8) at (xi , tn+1 ), 0 < γ < 1.

5 Numerical Experiments In this section, in order to solve the fractional Cauchy equation using the finitedifference discretization scheme (FDDS), the equation is presented in a discrete specific form. Anyhow, we consider four illustrated examples to demonstrate the performance and efficiency of the proposed algorithm. The computations are performed by Wolfram-Mathematica software 11. Example 1 Consider the following homogeneous fractional Cauchy equation: ∂ γ u(x, t) ∂ β u(x, t) +  = 0, ∂t γ ∂xβ

(18)

subject to the initial condition u(x, 0) = sin(π x),

(19)

where  = 1 × 10−3 , t ≥ 0, x ∈ [1, 4], γ = 1, and 0 < β ≤ 1. In particular, the exact solution of IVPs (18) and (19) at β = 1, γ = 1 is given by u(x, t) = sin(π (x − t)). Following the FDD algorithm, using h = 0.05 and

Solving Time-Space-Fractional Cauchy Problem …

35

τ = 0.01, the numerical results of FDDS with varying fractional order β such that β ∈ {0.75, 0.85, 0.95, 1}, γ = 1, compared with exact solution are given in Table 1 at the time t = 0.5 and x ∈ [1, 1.5]. In the light of showing the agreement between the FDDS and exact solutions, the absolute error of IVPs (18) and (19) are listed in Table 2 for β = 1, γ = 1 when t = 0.5 and x ∈ [1, 1.5] with h = 0.1. Table 3 is Table 1 Numerical results for Example 1 at t = 0.5, γ = 1 with varying β x

Exact

β=1

β = 0.95

β = 0.85

β = 0.75

1.05

−0.154883

−0.156431

−0.156431

−0.156432

−0.156432

1.10

−0.307523

−0.309013

−0.309014

−0.309014

−0.309015

1.15

−0.452590

−0.453987

−0.454008

−0.454031

−0.454036

1.20

−0.586514

−0.587782

−0.587832

−0.587886

−0.587900

1.25

−0.705995

−0.707104

−0.707186

−0.707276

−0.707301

1.30

−0.808093

−0.809015

−0.809130

−0.809257

−0.809294

1.35

−0.890292

−0.891005

−0.891152

−0.891317

−0.891366

1.40

−0.950570

−0.951055

−0.951232

−0.951433

−0.951495

1.45

−0.987441

−0.987688

−0.987892

−0.988124

−0.988199

1.50

−0.999999

−1.000000

−1.000230

−1.000490

−1.000570

Table 2 Absolute errors for Example 1 at β = 1, γ = 1

Table 3 FDDS of Example 1 at β = 0.95, γ = 1 with varying time T

x

Exact

FDDS

Absolute error

1.1

−0.307523

−0.309013

1.49065 × 10−3

1.2

−0.586514

−0.587782

1.26842 × 10−3

1.3

−0.808093

−0.809015

0.92203 × 10−3

1.4

−0.950570

−0.951055

0.48539 × 10−3

1.5

−0.999999

−1.000000

1.23369 × 10−6

x

T = 0.5

T = 1.0

1.05

−0.156431

−0.156428

1.10

−0.309014

−0.309010

1.15

−0.454008

−0.454027

1.20

−0.587832

−0.587880

1.25

−0.707186

−0.707267

1.30

−0.809130

−0.809245

1.35

−0.891152

−0.891300

1.40

−0.951232

−0.951412

1.45

−0.987892

−0.988099

1.50

−1.000230

−1.000460

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devoted to the FDDS approximate solutions at β = 0.95, γ = 1 with varying times t such that t = 0.5 and t = 1.0 over the interval [1, 1.5] with h = 0.05. From these tables, it can be noted that the FDDS approximate solutions are in good agreement with the exact solutions over the domain of interest. Anyhow, more iteration leads to more accurate solutions. For further analysis, the 2D-plot of the FDDS and exact solution for Example 1 are drawn in Fig. 1 at t = 0.5 and x ∈ [1, 3.5]. Whilst, the surface plot of the approximate solution at β = 0.95, γ = 1 is shown in Fig. 2. Example 2 Consider the following non-homogeneous fractional Cauchy equation: ∂ β u(x, t) ∂ γ u(x, t) +  = g(x, t), ∂t γ ∂xβ

(20)

subject to the initial condition u(x, 0) = x 2 sin(x),

(21)

where g(x, t) = sin(2x),  = 2, t ≥ 0, x ∈ [0, 0.9], β = 0.85, and 0 < γ ≤ 1. Fig. 1 FDDS and exact solutions for Example 1 at t = 0.5

Fig. 2 Surface plot of FDDS solution for Example 1 at β = 0.95, γ = 1

Solving Time-Space-Fractional Cauchy Problem …

37

Following the FDD algorithm, using h = 0.05 and τ = 0.025, the numerical results of FDDS with varying fractional order γ such that γ ∈ {0.75, 0.85, 0.95, 1}, β = 0.85 are given in Table 4 at the time t = 0.5 and x ∈ [0, 0.5]. Table 5 is devoted to the FDDS approximate solutions at β = 0.85 and γ = 0.95 with varying times t such that t = 0.25 and t = 0.5 over the interval [0, 0.4] with h = 0.05, the 2D-plot of the FDDS for Example 5.2 is drawn in Fig. 3 at t = 0.5 and x ∈ [0, 0.9]. Figure 4 shown the FDDS approximate solutions at β = 0.85 and γ = 0.95 with varying times t such that t = 0.25 and t = 0.5 over the interval [0, 0.4] . Whilst, the surface plot of the approximate solution at β = 0.85, γ = 1 is shown in Fig. 5 at t = 1.0. Example 3 Consider the following homogeneous fractional Cauchy equation: ∂ γ u(x, t) ∂ β u(x, t) +  = 0, ∂t γ ∂xβ

(22)

Table 4 Numerical results for Example 2 at t = 0.5, β = 0.85, with varying γ x

γ = 0.75

γ = 0.85

γ = 0.95

γ = 1.0

0.00

0.059580

0.054150

0.049026

0.046615

0.05

0.119865

0.108810

0.098405

0.093520

0.10

0.206253

0.183249

0.162686

0.153359

0.15

0.304010

0.266338

0.233562

0.218978

0.20

0.417262

0.360035

0.311871

0.290902

0.25

0.542894

0.462352

0.396428

0.368253

0.30

0.681071

0.573192

0.487204

0.451077

0.35

0.831202

0.692349

0.584309

0.539609

0.40

0.993351

0.820042

0.688155

0.634342

0.45

1.167720

0.956699

0.799340

0.735934

0.50

1.354800

1.102950

0.918616

0.845175

Table 5 FDDS of Example 2 at β = 0.85, γ = 0.95 with varying time T

x

T = 0.25

T = 0.5

0.0

0.024218

0.049026

0.05

0.048509

0.098405

0.1

0.076892

0.162686

0.15

0.107694

0.233562

0.2

0.141272

0.311871

0.25

0.177948

0.396428

0.3

0.218268

0.487204

0.35

0.262869

0.584309

0.4

0.312441

0.688155

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Fig. 3 FDDS for Example 2 at t = 0.5 with varying γ

Fig. 4 Behaviour of approximate solutions of Example 2 for different T

Fig. 5 Surface plot of FDDS solution for Example 2 at t = 1.0

subject to the initial condition   u(x, 0) = sin x 2 ,

(23)

where  = π , t ≥ 0, x ∈ [0, 0.95], β = 0.75, and 0 < γ ≤ 1. Following the FDD algorithm, using h = 0.05 and τ = 0.025, numerical results of FDDS with varying fractional order γ such that γ ∈ {0.75, 0.85, 0.95, 1}, β = 0.75 are given in Table 6 at the time t = 0.5 and x ∈ [0, 0.5].

Solving Time-Space-Fractional Cauchy Problem …

39

Table 6 Numerical results for Example 3 at t = 0.5, β = 0.75, with varying γ x

γ = 0.75

γ = 0.85

γ = 0.95

γ = 1.0

0.05

0.002266

0.002287

0.002307

0.002316

0.10

0.009407

0.009473

0.009533

0.009560

0.15

0.024289

0.024174

0.024053

0.023992

0.20

0.051221

0.050313

0.049423

0.048994

0.25

0.094644

0.091597

0.088738

0.087400

0.30

0.160305

0.152586

0.145614

0.142435

0.35

0.254323

0.237849

0.223484

0.217086

0.40

0.383737

0.352317

0.325831

0.314297

0.45

0.556090

0.500969

0.455986

0.436812

0.50

0.779532

0.688842

0.617112

0.587159

Table 7 FDDS of Example 3 at β = 0.75, γ = 0.95 with varying time T

x

T = 0.25

T = 0.5

0.05

0.0024045

0.0023072

0.10

0.0097809

0.0095331

0.15

0.0233115

0.0240534

0.20

0.0447471

0.0494235

0.25

0.0752926

0.0887387

0.30

0.1161080

0.1456140

0.35

0.1681450

0.2234840

0.40

0.2321930

0.3258310

Table 7 is devoted to the FDDS approximate solutions at β = 0.75 and γ = 0.95 with varying times t such that t = 0.25 and t = 0.5 over the interval [0, 0.4] with h = 0.05, the 2D-plot of the FDDS for Example 5.3 is drawn in Fig. 6 at t = 0.5 and x ∈ [0, 0.95]. Figure 7 shown the FDDS approximate solutions at β = 0.75 and γ = 0.95 with varying times t such that t = 0.25 and t = 0.5 over the interval Fig. 6 FDDS for Example 3 at β = 0.75, t = 0.5 with varying γ

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Fig. 7 FDDS for Example 3 at β = 0.75 and γ = 0.95 with varying times

Fig. 8 Surface plot of FDDS solution for Example 3, β = 0.75, γ = 1 at t = 1

[0, 0.4]. Whilst, the surface plot of the approximate solution at β = 0.75, γ = 1 is shown in Fig. 8 at t = 1.0. Example 4 Consider the following homogeneous fractional Cauchy equation: ∂ γ u(x, t) ∂ β u(x, t) +  = 0, ∂t γ ∂xβ

(24)

subject to the initial condition u(x, 0) = eξ x ,

(25)

where  = 0.1, ξ = 1.1771243444677, t ≥ 0, x ∈ [−2, 1], γ = 1 and 0 < β ≤ 1. In particular, the exact solution of IVPs (24) and (25) at β = 1 is given by u(x, t) = eξ (x−t) . Following the FDDS algorithm, using h = 0.0625 and τ = 0.01, the numerical results of the exact and FDDS for different values of fractional order β such that β ∈ {0.75, 0.85, 0.95, 1}, γ = 1 are given in Table 8 at the time t = 0.5 and x ∈ [−2, −1.25]. Table 9 is devoted to the FDDS approximate solutions at β = 0.95 with varying times t such that t = 0.5 and t = 1 over the interval [−2, −1.5] with h = 0.0625.

Solving Time-Space-Fractional Cauchy Problem …

41

Table 8 Numerical results for Example 4 at t = 0.5, γ = 1 with varying β x

Exact

β=1

β = 0.95

β = 0.85

β = 0.75

−2.0000

0.089537

0.094808

0.094822

0.094847

0.094868

−1.9375

0.096373

0.102192

0.102184

0.102175

0.102171

−1.8750

0.103730

0.109994

0.111023

0.112139

0.112442

−1.8125

0.111649

0.118391

0.119907

0.121624

0.122163

−1.7500

0.120173

0.127429

0.129290

0.131461

0.132197

−1.6875

0.129347

0.137158

0.139306

0.141862

0.142774

−1.6250

0.139222

0.147629

0.150043

0.152954

0.154028

−1.5625

0.149851

0.158900

0.161572

0.164827

0.166060

−1.5000

0.161291

0.171031

0.173965

0.177565

0.178954

−1.4375

0.173605

0.184088

0.187291

0.191244

0.192792

−1.3750

0.186859

0.198142

0.201627

0.205946

0.207658

−1.3125

0.201124

0.213269

0.217050

0.221754

0.223637

−1.2500

0.216479

0.229551

0.233647

0.238756

0.240817

Table 9 FDDS of Example 4 at β = 0.95 with varying time t

Fig. 9 Solution behaviour Example 4 for different values of β

x

t = 0.5

t=1

−2.0000

0.094822

0.094677

−1.9375

0.102184

0.102151

−1.8750

0.111023

0.112044

−1.8125

0.119907

0.121425

−1.7500

0.129290

0.131169

−1.6875

0.139306

0.141487

−1.6250

0.150043

0.152500

−1.5625

0.161572

0.164299

−1.5000

0.173965

0.176962

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Fig. 10 FDDS and exact solutions for Example 4 at t = 0.5

Figure 9 displays the approximate solutions of IVPs (24) and (25) for different values of fractional order β such that β ∈ {0.75, 0.85, 0.95, 1}, γ = 1 at time t = 0.5 and x ∈ [−2, 1]. The 2D-plot of the FDDS and exact solution for Example 5.4 are drawn in Fig. 10 at t = 0.5 and x ∈ [−2, 1]. Figure 11 shown the FDDS approximate solutions at β = 0.95 and γ = 1.0 with varying times t such that t = 0.5 and t = 1 over the interval [−2, 1] . Whilst, the surface plot of the FDDS approximate solution at β = 0.95, γ = 1 is shown in Fig. 12. From these graphs, it can be concluded that Fig. 11 Solution behaviour of Example 4 with different time-level T

Fig. 12 Surface plot of FDDS solution for Example 4 at β = 0.95

Solving Time-Space-Fractional Cauchy Problem …

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the behaviour of the FDDS approximate solutions are in good agreement with each other at different values of β.

6 Conclusion In this chapter, a new finite-difference technique has been developed for solving linear Cauchy equation of fractional order. We introduce the implicit scheme by discretization of the space-Riemann-Liouville fractional integral, and time-Caputo fractional derivative, the solution obtained using this technique shows that this approach can solve the problem effectively. The basic idea of this approach can be further utilised to resolve the linear Cauchy equation of fractional order with a variable coefficient or apply the finite volume method by using the same discretization. Funding This research was funded by Ajman University, UAE (Grant ID 2020-COVID 19-08: GL: 5211529). Acknowledgements The first author gratefully acknowledges support from Taibah University, Saudi Arabia, whilst the second author gratefully acknowledges support from Zarqa University, Jordan.

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On Modification of an Adaptive Stochastic Mirror Descent Algorithm for Convex Optimization Problems with Functional Constraints Mohammad S. Alkousa

Abstract This paper is devoted to a new modification of a recently proposed adaptive stochastic mirror descent algorithm for constrained convex optimization problems in the case of several convex functional constraints. Algorithms, standard and its proposed modification, are considered for the type of problems with non-smooth Lipschitz-continuous convex objective function and convex functional constraints. Both algorithms, with an accuracy ε of the approximate solution to the problem, are optimal in the terms of lower bounds of estimates and have the complexity  O ε−2 . In both algorithms, the precise first-order information, which connected with (sub)gradient of the objective function and functional constraints, is replaced with its unbiased stochastic estimates. This means that in each iteration, we can still use the value of the objective function and functional constraints at the research point, but instead of their (sub)gradient, we calculate their stochastic (sub)gradient. Due to the consideration of not all functional constraints on non-productive steps, the proposed modification allows saving the running time of the algorithm. Estimates for the rate of convergence of the proposed modified algorithm is obtained. The results of numerical experiments demonstrating the advantages and the efficient of the proposed modification for some examples are also given.

1 Introduction Large-scale non-smooth convex optimization is a common problem for a range of computational areas including statistics, computer vision, general inverse problems, machine learning, data science and in many applications arising in applied sciences and engineering. Since what matters most in practice is the overall computational time M. S. Alkousa (B) Moscow Institute of Physics and Technology (National Research University), Moscow, Russia e-mail: [email protected] National Research University, Higher School of Economics, Moscow, Russia © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 D. Zeidan et al. (eds.), Computational Mathematics and Applications, Forum for Interdisciplinary Mathematics, https://doi.org/10.1007/978-981-15-8498-5_3

47

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to solve the problem, first-order methods with computationally low-cost iterations become a viable choice for large- scale optimization problems. Generally, first-order methods have simple structures with a low memory requirement. Thanks to these features, they have received much attention during the past decade. There are a lot of first-order methods for solving the optimization problems in the case of non-smooth objective function. Some examples of these methods, to name but a few, are subgradient methods [23, 26, 30], subgradient projection methods [23, 26, 30], OSGA [22], bundle-level method [23], Lagrange multipliers method [9] and many others. There is a long history of studies on continuous optimization with functional constraints. The recent works on first-order methods for convex optimization with convex functional constraints include [5, 14, 16, 32–34] for deterministic constraints and [1, 2, 15, 35] for stochastic constraints. However, the parallel development for problems with non-convex objective functions and also with non-convex constraints, especially for theoretically provable algorithms, remains limited, see [17] and references therein. The mirror descent algorithm which originated in [20, 21] and was later analysed in [7], is considered as the non-Euclidean extension of subgradient methods. The standard subgradient methods employ the Euclidean distance function with a suitable step-size in the projection step. Mirror descent extends the standard projected subgradient methods by employing a non-linear distance function with an optimal step size in the non-linear projection step [18]. Mirror descent method not only generalises the standard gradient descent method, but also achieves a better convergence rate [12]. In addition, mirror descent method is applicable to optimization problems in Banach spaces where gradient descent is not [12]. An extension of the mirror descent method for constrained problems was proposed in [6, 20]. Usually, the step-size and stopping rule for mirror descent algorithms is required to know the Lipschitz constant of the objective function and constraint, if any. Adaptive step sizes, which do not require this information, are considered for unconstrained problems in [8], and for constrained problems in [6]. Some optimal mirror descent algorithms, for convex optimization problems with non-smooth convex functional constraint and both adaptive step sizes and stopping rules, are proposed in [5]. Also, there were considered some modifications of these algorithms for the case of problems with many functional constraints in [31]. If we focus on the problems of minimization of an objective function  consisting of a large number of component functionals, such as f (x) = Nj=1 fj (x) where fj : Rn → R, j = 1, N are convex, then in each iteration of any iterative  minimization procedure computing a single (sub)gradient ∇f (x) = Nj=1 ∇fj (x) becomes very expensive. Therefore, there is an incentive to calculate the stochastic (sub)gradient ∇f (x, ζ ), where ζ is a random variable taking its values in {1, . . . , N }. This means that ∇f (x, ζ ) = ∇fi (x), where i is chosen randomly in each iteration from the set {1, . . . , N }, or instead, one can employ randomly chosen a minibatch approach in which a small subset S ⊂ {1, . . . , N } is chosen randomly, then

On Modification of an Adaptive Stochastic Mirror Descent Algorithm …

49

 ∇f (x, ζ ) = i∈S ∇fi (x). This randomly calculating of the (sub)gradient is known as stochastic (sub)gradient. In the stochastic version of an optimization method, the exact first-order information is replaced with its unbiased stochastic estimates, where the exact first-order information is unavailable. This permits accelerating the solution process, with the earning from randomisation growing progressively with problem’s sizes. A different approach to solving stochastic optimization problems is called stochastic approximation (SA), which was initially proposed in a seminal paper by Robbins and Monro in 1951 [28]. An important improvement of this algorithm was developed by Polyak and Juditsky [24, 25]. More recently, Nemirovski et al. [19] presented a modified stochastic approximation method and demonstrated its superior numerical performance for solving a general class of non-smooth convex problems. This paper is devoted to a new modification of an adaptive stochastic mirror descent algorithm (see Algorithm 4 in [5]. This algorithm is listed as Algorithm 1, below, which is proposed to solve the stochastic setup (randomized version) of the convex minimization problems in the case of several convex functional constraints. This means that we can still use the value of the objective function and functional constraints at the research point, but instead of their (sub)gradient, we use their stochastic (sub)gradient. Namely, that we consider the first-order unbiased oracle that produces stochastic (sub)gradients of the objective function and functional constraints, see, for example, [13, 29]. We consider the arbitrary proximal structure and the type of problems with non-smooth Lipschitz-continuous objective function. Furthermore, it has been proved a theorem to estimate the rate of convergence of the proposed modification, from this theorem we can see that the modified algorithm achieves the optimal complexity of the order O ε−2 for the class of problems under consideration (see [20]). The rest of the paper is organised as follows. In Sect. 2, we give some basic notation, summarize the problem statement and standard mirror descent basics. In Sect. 3, we display the adaptive stochastic mirror descent algorithm (Algorithm 4 in [5]). Section 4 is devoted to the proposed modified algorithm and proving a theorem about the rate of convergence of this algorithm and its optimal complexity estimate. In the last section, we consider some numerical experiments that allow us to compare the work of the standard algorithm and its proposed modification for certain examples.

2 Problem Statement and Standard Mirror Descent Basics Let V be a finite-dimensional vector space, endowed with the norm  · , and V∗ is the conjugate space of V with the following norm: h∗ = max{h, x : x ≤ 1}, x

where h, x is the value of the continuous linear functional h at x ∈ V.

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Let Q ⊂ V be a closed convex set, f and gj : Q → R (j = 1, m) convex subdifferentiable functionals. We assume that f and gj (j = 1, m) are Lipschitz-continuous, i.e. there exist Mf > 0 and Mg > 0, such that |f (x) − f (y)| ≤ Mf x − y ∀ x, y ∈ Q,

(1)

|gj (x) − gj (y)| ≤ Mg x − y ∀ x, y ∈ Q, j = 1, m.

(2)

It is clear that instead of a set of functionals {gj (·)}m j=1 we can see one functional g : Q → R, such that g(x) = max {gj (x)}, |g(x) − g(y)| ≤ Mg x − y

∀ x, y ∈ Q.

j=1,m

It means that at every point x ∈ Q there is a subgradient ∇g(x), and ∇g(x)∗ ≤ Mg . Recall that for a differentiable functional g, the subgradient ∇g(x) coincides with the usual gradient. In this paper, we consider the stochastic setup of the following convex constrained optimization problem: (3) f (x) → min . x∈Q, g(x)≤0

For the stochastic setup of the problem (3), we introduce the following assumptions (see [4, 5]). Given a point x ∈ Q, we can calculate the stochastic (sub)gradients ∇f (x, ξ ) and ∇g(x, ζ ), where ξ and ζ are random vectors. These stochastic (sub)gradients satisfy E[∇f (x, ξ )] = ∇f (x) ∈ ∂f (x) and E[∇g(x, ζ )] = ∇g(x) ∈ ∂g(x),

(4)

where E denote the expectation, and ∇f (x, ξ )∗ ≤ Mf and ∇g(x, ζ )∗ ≤ Mg ,  To motivate these assumptions, let Sn (1) = x ∈

Rn+

|

a.s. in ξ, ζ. n 

(5)

 xi = 1 be a standard

i=1

unit simplex in Rn , we consider the following optimization problem: ⎧ ⎨f (x) = 21 Ax, x → min , x∈Sn (1)

⎩s.t. g(x) = max {ci , x } ≤ 0, i=1,m

where A is a given n × n matrix and ci (i = 1, m) are given vectors in Rn (see [5]).

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The exact computation of the gradient ∇f (x) = Ax takes O(n2 ) arithmetic operations, which is expensive, when n is very large, for the huge-scale optimization problems. In this setting, it is natural to use the randomisation to construct a stochastic approximation for ∇f (x). Let ξ be a random variable its values 1, . . . , n with probabilities x1 , . . . , xn , respectively. Let Ai denote the ith column of the matrix A. Since x ∈ Sn (1), E[Aξ ] = A1 P(ξ = 1) + · · · + An P(ξ = n)     x1

xn

1

n

= A x1 + · · · + A xn = Ax, where P denote to the probability of an event. Thus, we can use Aξ as a stochastic gradient of f (i.e. ∇f (x, ξ ) = Aξ ), which can be calculated in O(n) arithmetic operations. Let d : Q → R be a distance-generating function, which is continuously differentiable and 1-strongly convex with respect to the norm  · , i.e., 1 d (y) ≥ d (x) + ∇d (x), y − x + y − x2 ∀ x, y ∈ Q, 2 and assume that min d (x) = d (0). Suppose, we have a constant 0 > 0 such that x∈Q

d (x∗ ) ≤ 20 , where x∗ is a solution to the problem (3). Note that if there is a set of optimal points for (3) X∗ ⊂ Q, we may assume that min d (x∗ ) ≤ 20 .

x∗ ∈X∗

For all x, y ∈ Q ⊂ V, we consider the corresponding Bregman divergence, which was initially studied by Bregman [10] and later by many others (see [3]), Vx (y) = d (y) − d (x) − ∇d (x), y − x . In particular, in the standard proximal setup (i.e. Euclidean setup), we can choose d (x) = 21 x22 , then Vx (y) = 21 x − y22 . Another setup, for example, entropy, 1 /2 , simplex, spectrahedron and many others, can be found in [8]. We also assume that the constant 0 > 0 is known, such that sup Vx (y) ≤ 20 .

(6)

x,y∈Q

For all x ∈ Q and p ∈ V∗ , the proximal mapping operator (mirror descent step) is defined as   Mirr x (p) = arg min p, u + Vx (u) . u∈Q

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We make the simplicity assumption, which means that Mirr x (p) is easily computable. Let x∗ be a solution to (3) and ε > 0 is given, we say that a (random) point xˆ ∈ Q is an expected ε–solution to (3) if E[f (ˆx)] − f (x∗ ) ≤ ε and g(ˆx) ≤ ε.

(7)

The following well-known lemma describes the main property of the proximal mapping operator (see [5, 8]). Lemma 1 Let f : Q → R be a convex subdifferentiable function over the convex set Q and z = Mirry (h∇f (y, ξ )) for some h > 0, y, z ∈ Q and ξ random vector. Then for each x ∈ Q we have h (f (y) − f (x)) ≤

h2 ∇f (y, ξ )2∗ + Vy (x) − Vz (x) + h ∇f (y, ξ ) − ∇f (y), y − x . 2

3 Adaptive Stochastic Mirror Descent Algorithm In [5], it was considered an adaptive method, for the convex optimization problem (3) in the stochastic setup described above (see Algorithm 1, below). In this setting, the output of the algorithm is random, in the sense of (7). The adaptivity of this method is in terms of step-size and stopping role, which means that we do not need to know the constants Mf and Mg in advance. We assume that, on each iteration of the algorithm, independent realisations of the random variables ξ and ζ are generated. In this section, we show this algorithm and the fundamental result of the estimate about the convergence rate of this algorithm. As can be seen from the items of the Algorithm 1, the needed point (Ensure) is selected among the points xk for which g(xk ) ≤ ε. Therefore, we will call step k productive if g(xk ) ≤ ε. If the reverse inequality g(xk ) > ε holds, then step k will be called non-productive. Let I , J denote the set of indexes of productive and non-productive steps produced by Algorithm 1, respectively. NI , NJ denote the number of productive and nonproductive steps, respectively. For the complexity estimate of Algorithm 1, the next result was obtained in [4, 5]. Theorem 1 Let equalities (4) and inequalities (5) hold. Assume that a known constant 0 > 0 is such that inequality (6) holds. Then Algorithm 1 stops after no more than

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53

Algorithm 1 Adaptive stochastic Mirror Descent algorithm: objective function is Lipschitz continuous. Require: accuracy ε, starting point x0 , d (·), Q, 0 such that (6) holds. 1: I =: ∅ 2: N ← 0 3: repeat 4: if g(xN ) ≤ ε then  5: MN := ∇f (xN , ξ N )∗ , N −1/2  2 6: hN = 0 Mt , t=0   N +1 := MirrxN hN ∇f (xN , ξ N ) , "productive step" 7: x 8: N →I 9: else   10: MN := ∇g(xN , ζ N )∗ , N −1/2  2 11: hN = 0 Mt , t=0   12: xN +1 := MirrxN hN ∇g(xN , ζ N ) , "non-productive step" 13: end if 14: N ← N + 1 N −1 1/2  2 15: until N ≥ 2ε 0 Mt . t=0  1 Ensure: x¯ N := NI xk . k∈I

 N=

4 max{Mf2 , Mg2 }20



ε2

(8)

iterations and x¯ N is an expected ε–solution to problem (3) in the sense of (7).

4 The Modification of an Adaptive Stochastic Mirror Descent Algorithm In this section, we consider a modification of an Algorithm 1. The idea of this modification was considered in [31] for some adaptive mirror descent algorithms to solve the deterministic setup of the convex optimization problems with Lipschitz-continuous functional constraints. This idea is summarised as: when we have a non-productive step k, i.e. g(xk ) > ε, then instead of calculating the subgradient of the functional constraint with max-type g(x) = max {gi (x)}, we calculate (sub)gradient of one funci=1,m

tional gj , for which we have gj (xk ) > ε. The proposed modification allows saving the running time of algorithm due to consideration of not all functional constraints on non-productive steps.

54

M. S. Alkousa

Algorithm 2 The Modification of an adaptive stochastic Mirror Descent: objective function is Lipschitz-continuous. Require: accuracy ε, starting point x0 , d (·), Q, 0 such that (6) holds. 1: I =: ∅ 2: N ← 0 3: repeat 4: if g(xN ) ≤ ε then  5: MN := ∇f (xN , ξ N )∗ , N −1/2  2 6: hN = 0 Mt , t=0   N +1 := MirrxN hN ∇f (xN , ξ N ) , "productive step" 7: x 8: N →I 9: else 10: (i.e. gj(N) (xN ) > ε for some  j(N ) ∈ {1, . . . , m}) 11: MN := ∇gj(N ) (xN , ζ N )∗ , N −1/2  2 Mt , 12: hN = 0 t=0   13: xN +1 := MirrxN hN ∇gj(N ) (xN , ζ N ) , "non-productive step" 14: end if 15: N ← N + 1 N −1 1/2  2 16: until N ≥ 2ε 0 Mt . t=0 1  k N x . Ensure: x¯ := NI k∈I



Denote δk =

∇f (xk , ξ k ) − ∇f (xk ), xk − x∗ if k ∈ I , ∇g(xk , ζ k ) − ∇g(xk ), xk − x∗ if k ∈ J .

By Lemma 1, with y = xk , z = xk+1 and x = x∗ , we have for all k ∈ I f (xk ) − f (x∗ ) ≤

 hk  ∇f (xk , ξ k )2 + Vxk (x∗ ) − Vxk+1 (x∗ ) + ∗ 2 hk hk + ∇f (xk , ξ k ) − ∇f (xk ), xk − x∗ ,

(9)

the same for all k ∈ J , we have (remember that, with gj(k) (·) we mean any constraint, such that gj(k) (xk ) > ε), gj(k) (xk ) − gj(k) (x∗ ) ≤

hk Vxk (x∗ ) Vxk+1 (x∗ ) ∇gj(k) (xk , ζ k )2∗ + − + 2 hk hk + ∇gj(k) (xk , ζ k ) − ∇gj(k) (xk ), xk − x∗ .

(10)

Taking summation, in each side of (9) and (10), over productive and nonproductive steps, we get

On Modification of an Adaptive Stochastic Mirror Descent Algorithm …

55

−1     N hk Mk2 + f (xk ) − f (x∗ ) + gj(k) (xk ) − gj(k) (x∗ ) ≤ 2 k∈I

k∈J

k=0

+

N −1  k=0

N −1

 1 δk . (Vxk (x∗ ) − Vxk+1 (x∗ )) + hk k=0

Using (6), we have N −1  k=0

 1 1 Vxk (x∗ ) − Vxk+1 (x∗ ) = Vx0 (x∗ )+ hk h0  N −2   1 1 1 Vxk+1 (x∗ ) − + − Vxk (x∗ ) ≤ hk+1 hk hN −1 k=0

 1 20 20 1 = + 20 − . h0 hk+1 hk hN −1 N −2

≤

k=0

Whence, by the definition of step-sizes hk −1     N Mk2 0 f (xk ) − f (x∗ ) + gj(k) (xk ) − gj(k) (x∗ ) ≤  1/2 + 2 k 2 k∈I k∈J k=0 M i i=0 N −1 1/2 N −1   + 0 Mk2 + δk ≤ k=0

≤ 20

N −1 

k=0

1/2 Mk2

+

k=0

N −1 

δk ,

k=0

(11) where we used the inequality N −1  k=0



Mk2

k i=0

Mi2

1/2 ≤ 2

N −1 

1/2 Mk2

,

k=0

which can be proved by induction. Since, for k ∈ J , gj(k) (xk ) − gj(k) (x∗ ) ≥ gj(k) (xk ) > ε, we get    gj(k) (xk ) − gj(k) (x∗ ) > ε = εNJ . k∈J

k∈J

Thus from (11) and the stopping criterion of Algorithm 2, we have

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M. S. Alkousa





f (x ) − f (x∗ ) < 20 k

N −1 

k∈I

1/2 +

Mk2

k=0

N −1 

δk − εNJ

(12)

δk

(13)

k=0

≤ ε(NI + NJ ) − εNJ +

N −1  k=0

= εNI +

N −1 

δk .

(14)

k=0

We can rewrite (14) as follows: 

f (xk ) − NI f (x∗ ) < εNI +

k∈I

N −1 

δk .

(15)

k=0

By the convexity of f , we get   NI f

1  k x NI



 −f

∗

< εNI +

k∈I

N −1 

δk ,

(16)

k=0

where f ∗ = f (x∗ ). By the definition of x¯ N (see the Ensure of Algorithm 2), we get the following inequality: N −1    N  ∗ NI f x¯ − f < εNI + δk .

(17)

k=0

As long as the inequality (17) is strict, the case of I = ∅ is impossible (i.e. NI = 0). Now by taking the expectation in (17), we obtain  N −1      δk , E f x¯ N − f (x∗ ) ≤ ε + E NI k=0

but

N −1 k=0

E

δk NI

!

= 0, (see [4]). Thus    E f x¯ N − f (x∗ ) ≤ ε.

(18)

At the same time, for k ∈ I it holds that g(xk ) ≤ ε. Then, by the definition of x¯ N and the convexity of g we get   1  g(xk ) ≤ ε. g x¯ N ≤ NI k∈I

On Modification of an Adaptive Stochastic Mirror Descent Algorithm …

57

Thus we have come to the following result. Theorem 2 Let equalities (4) and inequalities (5) hold. Assume that a known constant 0 > 0 is such that inequality (6) holds. Then Algorithm 2 stops after no more than   4 max{Mf2 , Mg2 }20 N= (19) ε2 iterations and x¯ N is an expected ε–solution to problem (3) in the sense of (7). Remark 1 From the estimate  (19), we can see that Algorithm 2 achieves the complexity of the order O ε−2 , which is an optimal, for the studied class of non-smooth functions, from the point of view of the theory of lower bounds of estimates, according to Nemirovski and Yudin (see [20]).

5 Numerical Experiments In order to compare Algorithms 1 and 2, and to show the advantages of the proposed modified algorithm some numerical tests were carried out. We consider some different examples of the following non-smooth finite-sum problem:  N 1  min f (x) := fi (x) , x N i=1 

(20)

where each summand fi is a Lipschitz-continuous function. This problem is ubiquitous in many areas and applications, in particular in machine learning applications, f is the total loss function whereas each fi represents the loss due to the ith training sample [11, 27]. In our experiments, we consider the following two examples of the problem (20): Example 1 f (x) =

N 1  |ai , x − bi | , N i=1

where the coefficients ai ∈ Rn and bi ∈ R for each i = 1, . . . , N . Example 2 f (x) =

N 1  0.5Ci x, x , N i=1

where Ci ∈ Rn×n , for each i = 1, . . . , N , are positive-definite matrices, i.e. Ci  0.

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For the coefficients ai ∈ Rn and constants bi ∈ R (i = 1, . . . , N ), in example 1, with different values of N . Let A ∈ RN ×(n+1) be a matrix with entries drawn from different random distributions. Then aiT are rows in the matrix A ∈ RN ×n , which is obtained from A, by eliminating the last column, and bi are the entries of the last column in the matrix A. The positive-definite matrices Ci  0 (i = 1, . . . , N ), in Example 2, with different values of N , are drawn from different random distributions. In more details, the entries of A and Ci (i = 1, . . . , N ), with different values of N , are drawn 1. When N = 75, from the Gumbel distribution with the location of the mode equaling 1 and the scale parameter equaling 2. 2. When N = 100, from the standard exponential distribution with a scale parameter of 1. 3. When N = 150, from the uniform distribution over [0, 1). For the functional constraint g(x) = max{gi (x)}, we take m = 50, n = 1500 and i∈1,m

gi (x) = αi , x + βi linear functionals, where the coefficients αi ∈ Rn and βi ∈ R for i = 1, . . . , m are taken as follows: Let B ∈ Rm×(n+1) be a Toeplitz matrix with the first row (1, 1, . . . , 1) ∈ Rn+1 and the first column (1, 2, . . . , m)T . Then αiT are rows in the matrix B ∈ Rm×n , which is obtained from B, by eliminating the last column, and βi are the entries of the last column in the matrix B, i.e. the eliminated column. For more clarification, when m = 10 and n = 14, then the Toeplitz matrix B with the first row (1, 1, . . . , 1) ∈ R15 and the first column (1, 2, . . . , 10)T has the form ⎛

1 ⎜ 2 ⎜ ⎜ 3 ⎜ ⎜ 4 ⎜ ⎜ 5 B=⎜ ⎜ 6 ⎜ ⎜ 7 ⎜ ⎜ 8 ⎜ ⎝ 9 10

1 1 2 3 4 5 6 7 8 9

11111 11111 11111 21111 32111 43211 54321 65432 76543 87654

⎞ 11111111 1 1 1 1 1 1 1 1⎟ ⎟ 1 1 1 1 1 1 1 1⎟ ⎟ 1 1 1 1 1 1 1 1⎟ ⎟ 1 1 1 1 1 1 1 1⎟ ⎟ ∈ R10×15 . 1 1 1 1 1 1 1 1⎟ ⎟ 1 1 1 1 1 1 1 1⎟ ⎟ 1 1 1 1 1 1 1 1⎟ ⎟ 2 1 1 1 1 1 1 1⎠ 32111111

The proximal structure is given by Euclidean norm norm as   and squared Euclidean 1 √1 1 0 n √ √ a prox-function. We choose starting point x = , n , . . . , n ∈ R , ε = 0.05, n and Q = {x = (x1 , x2 , . . . , xn ) ∈ Rn | x12 + x22 + . . . + xn2 ≤ 1}. For any x = (x1 , . . . , xn ) and y = (y1 , . . . , yn ) in Q, the following inequality holds: 1 1 x − y22 = (xk − yk )2 ≤ x12 + . . . + xn2 + y12 + . . . + yn2 ≤ 2. 2 2 n

k=1

Therefore, we can choose 0 =

√ 2.

On Modification of an Adaptive Stochastic Mirror Descent Algorithm … Table 1 Results of Algorithms 1 and 2, for Examples 1 and 2, in R1500 N Example 1 Algorithm 1 Algorithm 2 Iterations Time (sec) Iterations 75 100 150 75 100 150

30 157 12 827 7 452 Example 2 104 513 18 814 5 451

59

Time (sec)

618.79 254.34 139.99

27 007 11 071 5 713

22.47 10.04 4.62

2008.12 358.02 115.47

90 154 17 584 4 834

82.38 15.3 5.45

Our experiments are motivated by the need to solve the problem (3) when either the dimension n is large or when the objective function f is of a finite-sum structure, as in examples 1 and 2, with N , the number of components, being large. We run Algorithms 1 and 2, in order to both Examples 1 and 2, with m = 50, n = 1500. The results of the work of Algorithms 1 and 2 are represented in Table 1, below. These results demonstrate the comparison between the number of iterations and the running time (in seconds) for each algorithm. All experiments were implemented in Python 3.4, on a computer fitted with Intel(R) Core(TM) i7-8550U CPU @ 1.80GHz, 1992 Mhz, 4 Core(s), 8 Logical Processor(s). RAM of the computer is 8GB. From Table 1, in order to both Examples 1 and 2, we can see that the modified Algorithm 2 always works better than Algorithm 1. It is clearly shown in all experiments according to the number of iterations and especially according to the running time of the algorithms. The running time of Algorithm 2 is very small compared to the running time of Algorithm 1 (on average, it is smaller 25 times). This feature of the Algorithm 2 is very important in all applications of mathematical optimization. Remark 2 Now, as in the previous, to compare Algorithms 1 and 2, with m = 50, n = 100 and different values of N , some additional numerical tests were carried out. The coefficients αi ∈ Rn and βi ∈ R, for each i = 1, . . . , m, are the entries of the Toeplitz matrix, which is described above. The entries of the matrices Ci (i = 1, . . . , N ) are drawn from the uniform distribution over [0, 1). √ We run Algorithms 1 and 2 with the same previous parameters ε = 0.05, 0 = 2 and the set Q. The results of Algorithms 1 and 2, in order to the Examples 1 and 2, are represented in Table 2. These results demonstrate the comparison between the number of iterations and the running time (in seconds) for each algorithm, with different values of N . From Table 2, we can see that Algorithm 2 works better than Algorithm 1 according to the number of iterations and especially according to the running time of algorithms.

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Table 2 The results of Algorithms 1 and 2, for Examples 1 and 2, with different values of N N Example 1 Algorithm 1 Algorithm 2 Iterations Time (sec) Iterations Time (sec) 1 000 5 000 10 000 15 000 25 000 50 000 75 000 100 000 125 000 150 000

6 717 5 726 8 017 6 427 6 775 7 339 6 599 6 235 6 709 6 928 Example 2 6 519 6 238 5 364 5 862 6 025 5 341 6 227 5 847 5 486 6 294 6 055

1 000 2 500 5 000 7 500 10 000 12 500 15 000 17 500 20 000 22 500 25 000

9.770 7.975 11.076 8.890 9.530 10.232 9.160 8.665 9.175 9.671

5 366 5 334 5 574 5 243 5 348 6 187 5 287 5 400 6 095 5 360

0.476 0.452 0.500 0.445 0.474 0.582 0.452 0.456 0.512 0.471

10.496 9.750 8.287 9.255 9.331 10.687 12.981 9.509 8.515 10.140 11.598

5 178 4 634 4 615 5 029 4 506 4 688 4 995 4 616 4 760 4 551 4 534

0.656 0.523 0.679 0.677 0.569 0.672 0.576 0.603 0.620 0.585 0.596

5.1 Additional Experiments: Fermat-Torricelli-Steiner Problem In this subsection, some additional numerical experiments connected with the analogue of the well-known Fermat-Torricelli-Steiner problem with some non-smooth functional constraints were carried out. For a given set {Ak = (a1k , a2k , . . . , ank ); k = 1, N } of N points, in n-dimensional Euclidean space Rn , we need to solve the problem (3), where the objective function f is given by f (x) :=

N (  k=1

(x1 − a1k )2 + (x2 − a2k )2 + . . . + (xn − ank )2 =

N  k=1

x − Ak 2 .

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Fig. 1 The results of Algorithms 1 and 2, for Fermat-Torricelli-Steiner problem, with m = 250, n = 1000, N = 100 and different values of ε

The functional constraint is given by g(x) = max{gi (x) = αi , x + βi }, where i∈1,m

the coefficients αi ∈ Rn and βi ∈ R are taken as in the previous experiments (the entries of the Toeplitz matrix B). We take the points Ak (k ∈ 1, N ) in the unit ball Q. The coordinates of these points are drawn from the uniform distribution over [0, 1). 0 n We choose √ the standard Euclidean proximal setup, starting point x = 0 ∈ R and 0 = 2. We run Algorithms 1 and 2 with n = 1000, m = 250, N = 100 and different values of accuracy ε ∈ {1/2i : i = 1, 2, 3, 4, 5, 6}. The results of the work of Algorithms 1 and 2, are presented in Fig. 1, left (the number of iterations produced by the studied algorithms to reach an ε–solution of the proposed problem as a function of accuracy) and Fig. 1, right (the required running time of the studied algorithms, in seconds, as a function of accuracy). From Fig. 1, we see that both 1 and 2 are optimal, where they achieve   Algorithms the complexity of the order O ε−2 , which is optimal estimate for the studied class of non-smooth functions. But Algorithm 2 is more efficiently and works better than Algorithm 1, according to the number of iterations and the running time. We note that the running time of Algorithm 1 is very long compared with the running time of Algorithm 2, where by Algorithm 2 one needs a few seconds to reach its stopping criterion and to achieve a solution of the problem, while it takes more minutes by Algorithm 1. Therefore, the efficiency of Algorithm 2 is represented by its very high execution speed compared with Algorithm 1.

6 Conclusions In this work, a new modification of an adaptive stochastic mirror descent algorithm was proposed to solve the stochastic setting of the convex minimization problem in the case of Lipschitz-continuous objective function and several convex functional constraints. In each iteration of the proposed modified algorithm, we calculate the

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stochastic (sub)gradient of the objective function or the functional of constraint, which is prevalent and effective in Machine Learning scenarios, large-scale optimization problems, and their applications. The proposed modification allows saving the running time of algorithm due to the consideration of not all functional constraints on non-productive steps. Furthermore, it has been proved a theorem to estimate the rate of convergence of the proposed modified algorithm. Numerical experiments for a geometrical problem, Fermat-Torricelli-Steiner problem, with convex constraints are presented. The results of carried out numerical experiments illustrate the advantages of the modified Algorithm 2 and illustrate that the running time of this Algorithm is very small compared to the running time of the standard Algorithm 1. Acknowledgements The author is very grateful to Alexander V. Gasnikov, Fedor S. Stonyakin and Alexander G. Biryukov for fruitful discussions.

References 1. Alkousa, M.S.: On some stochastic mirror descent methods for constrained online optimization problems. Comput. Res. Model. 11(2), 205–217 (2019) 2. Basu, K., Nandy, P.: Optimal Convergence for Stochastic Optimization with Multiple Expectation Constraints (2019). Available via DIALOG. https://arxiv.org/pdf/1906.03401.pdf 3. Bauschke, H.H., Borwein, J.M., Combettes, P.L.: Bregman monotone optimization algorithms. SIAM J. Control Opti. 42(2), 596–636 (2003) 4. Bayandina, A.: Adaptive Stochastic Mirror Descent for Constrained Optimization. 2017 Constructive Nonsmooth Analysis and Related Topics (dedicated to the memory of V.F. Demyanov) (CNSA), St. Petersburg. pp. 1–4 (2017) 5. Bayandina, A., Dvurechensky, P., Gasnikov, A., Stonyakin, F., Titov, A.: Mirror descent and convex optimization problems with non-smooth inequality constraints. In: Large-Scale and Distributed Optimization, pp. 181–213. Springer, Cham (2018) 6. Beck, A., Ben-Tal, A., Guttmann-Beck, N., Tetruashvili, L.: The comirror algorithm for solving nonsmooth constrained convex problems. Oper. Res. Lett. 38(6), 493–498 (2010) 7. Beck, A., Teboulle, M.: Mirror descent and nonlinear projected subgradient methods for convex optimization. Oper. Res. Lett. 31(3), 167–175 (2003) 8. Ben-Tal, A., Nemirovski, A.: Lectures on Modern Convex Optimization. Society for Industrial and Applied Mathematics, Philadelphia (2001) 9. Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, New York (2004) 10. Bregman, L.M.: The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming. USSR Comput. Mathemat. Mathemat. Phys. 7(3), 200–217 (1967) 11. Ding, Z., Chen, Y., Li, Q., Zhu, X.: Error Lower Bounds of Constant Step-size Stochastic Gradient Descent (2019). Available via DIALOG. https://arxiv.org/pdf/1910.08212.pdf 12. Doan, T.T., Bose, S., Nguyen, D.H., Beck, C.L.: Convergence of the iterates in mirror descent methods. IEEE Control Syst. Lett. 3(1), 114–119 (2019) 13. Duchi, J.C.: Introductory Lectures on Stochastic Convex Optimization. Park City Mathematics Institute, Graduate Summer School Lectures (2016) 14. Fercoq, O., Alacaoglu, A., Necoara, I., Cevher, V.: Almost Surely Constrained Convex Optimization (2019). Available via DIALOG. https://arxiv.org/pdf/1902.00126.pdf 15. Lan, G., Zhou, Z.: Algorithms for Stochastic Optimization with Functional or Expectation Constraints. Comput. Optim. Appl. 76, 461–498 (2020)

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16. Lin, Q., Ma, R., Yang, T.: Level-set methods for finite-sum constrained convex optimization. In: International Conference on Machine Learning, vol. 3118–3127 (2018) 17. Lin, Q., Ma, R., Xu, Y.: Inexact Proximal-Point Penalty Methods for Non-Convex Optimization with Non-Convex Constraints (2019). Available via DIALOG. https://arxiv.org/pdf/1908. 11518.pdf 18. Luong, D.V.N., Parpas, P., Rueckert, D., Rustem, B.: A weighted mirror descent algorithm for nonsmooth convex optimization problem. J. Opti. Theory Appl. 170(3), 900–915 (2016) 19. Nemirovski, A., Juditsky, A., Guanghui, L., Shapiro, A.: Robust stochastic approximation approach to stochastic programming. SIAM J. Opti. 19(4), 1574–1609 (2009) 20. Nemirovsky, A., Yudin, D.: Problem Complexity and Method Efficiency in Optimization. Wiley, New York (1983) 21. Nemirovskii, A.: Efficient methods for large-scale convex optimization problems. Ekonomika i Matematicheskie Metody (1979) (in Russian) 22. Neumaier, A.: OSGA: a fast subgradient algorithm with optimal complexity. Mathemat. Program. 158(1–2), 1–21 (2016) 23. Nesterov, Y.: Introductory Lectures on Convex Optimization. Springer Optimization and Its Applications, vol. 137 (2018) 24. Polyak, B.T., Juditsky, A.B.: Acceleration of stochastic approximation by averaging. SIAM J. Control Opti. 30(4), 838–855 (1992) 25. Polyak, B.T.: New stochastic approximation type procedures. Automat. i Telemekh. 51(7), 937–1008 (1990) 26. Polyak, B.: Introduction to Optimization. Optimization Software Inc., Publications Division, New York (1987) 27. Qian, X., Sailanbayev, A., Mishchenko, K., Richtárik, P.: MISO is Making a Comeback With Better Proofs and Rates (2019). Available via DIALOG. https://arxiv.org/pdf/1906.01474.pdf 28. Robbins, H., Monro, S.: A stochastic approximation method. Ann. Mathemat. Stat. 22(3), 400–407 (1951) 29. Shapiro, A., Dentcheva, D., Ruszczynski, A.: Lectures on Stochastic Programming: Modeling and Theory. Society for Industrial and Applied Mathematics, Philadelphia, PA (2014) 30. Shor, N.Z.: Minimization Methods for Non-differentiable Functions. Springer Series in Computational Mathematics, Springer (1985) 31. Stonyakin, F.S., Alkousa, M.S., Stepanov, A.N., Barinov, M.A.: Adaptive mirror descent algorithms in convex programming problems with Lipschitz constraints. Trudy Instituta Matematiki i Mekhaniki URO RAN 24(2), 266–279 (2018) 32. Titov, A.A., Stonyakin, F.S., Gasnikov, A.V., Alkousa, M.S.: Mirror descent and constrained online optimization problems. Optimization and applications. In: 9th International Conference OPTIMA-2018 (Petrovac, Montenegro, October 1–5, 2018). Revised Selected Papers. Communications in Computer and Information Science, vol. 974, pp. 64–78 (2019) 33. Wei, X., Yu, H., Ling, Q., Neely, M.: Solving non-smooth constrained programs with lower complexity than O(1/ε): A primal-dual homotopy smoothing approach. In: Advances in Neural Information Processing Systems, vol. 3995–4005 (2018) 34. Xu, Y.: Iteration complexity of inexact augmented lagrangian methods for constrained convex programming. Mathemat. Program. Series A 1–46 (2019) 35. Xu, Y.: Primal-Dual Stochastic Gradient Method For Convex Programs with Many Functional Constraints (2019). Available via DIALOG. https://arxiv.org/pdf/1802.02724.pdf

Inductive Description of Quadratic Lie and Pseudo-Euclidean Jordan Triple Systems Amir Baklouti and Samiha Hidri

Abstract This chapter is the first investigation of the notion of double extension to triple systems. We appropriate this notion of double extension to quadratic Lie triple systems so that we give an inductive description of all quadratic Lie triple systems. Moreover, we prove that any Jordan triple system is either a T ∗ -extension of a Jordan triple system or an ideal of codimension one of a T ∗ -extension. Many other results about Lie and Jordan triple systems are offered.

1 Introduction Lie triple systems arose initially in Cartan’s study of Riemannian geometry. Jacobson [18] first introduced them in connection with problems from Jordan theory and quantum mechanics, viewing Lie triple systems as subspaces of Lie algebras that are closely related to the ternary product. Lister gave the structure theory of Lie triple systems of characteristic 0 in [26]. Hopkin introduced the concepts of nilpotent ideals and the nil-radical of Lie triple systems, she successfully generalised Engel’s theorem to Lie triple systems in characteristic zero [17]. More recently, Lie triple systems have been connected with the study of the Yang-Baxter equations [21]. The first examples of Lie triple systems arise from Lie algebras by taking the triple product. If the Lie algebra is quadratic, then the Lie triple system arising is also quadratic. See [30]. Quadratic Lie algebra is studied and described inductively using A. Baklouti (B) Department of Mathematics, College of First Common Year, Umm Al-Qura University, P.O. Box 14035, Mecca 21955, Saudi Arabia e-mail: [email protected] Department of Mathematics, University of Sfax, IPEIS, Road of Menzel Chaker Km 0.5, 3000 Sfax, Tunisia S. Hidri Department of Mathematics, Faculty of Science, University of Sfax, Road of Soukra Km 3, B.P. 1171, 3000 Sfax, Tunisia © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 D. Zeidan et al. (eds.), Computational Mathematics and Applications, Forum for Interdisciplinary Mathematics, https://doi.org/10.1007/978-981-15-8498-5_4

65

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the double extension in the work of Medina and Revoy [29]. After that, this notion of double extension is used and generalised for many structures of algebras and superalgebras. See, for example, [1, 2, 4–7, 9, 10, 12, 14, 23]. In this work, we investigate how we can extend this notion to triple systems. We seek to understand the structure of triple system viewing as a double extension and then demonstrate that a non-simple quadratic (resp. pseud-Euclidean) Lie (resp. Jordan) triple system is a double extension of another one. After that, we prove that it can be reconstructed by double extension and/or direct sum of Lie triple systems of dimension one. This investigation needs many new definitions and new results as a preparation for the main theorem. Note that if we will mention some results getting from another references we will recall it without proof and precise this by some sentences before citing the recalling definition or result us: Now we will recall the following result. See, for example, Theorem 4. Definitions or results enunciated without the above precision, mean that these definitions or results are new and are ours. The first examples of quadratic (resp. pseud-Euclidean) Lie (resp. Jordan) are the semi-simple ones. The theory of these structures is developed in [6, 11, 19, 20, 27, 28, 30, 31]. After some preparations, in the preamble section, we show, in the first subsection of the third section that there exist a pseudo-Euclidean Jordan triple systems which is not semi-simple. We proceed for that by T ∗ -extension. This notion of T ∗ -extension was introduced by M. Bordemann, who proved in [13] that every Jordan algebra of even dimension n, which contains an isotropic ideal of dimension 2n , is a T ∗ extension of a Jordan algebra and every odd-dimensional Jordan algebra is an ideal of codimension one of a T ∗ -extension. J. Lin and all extended this notion in [25] to Lie triple systems. We will show that the same theorem holds for Jordan triple systems. The proof in this case is different from the Jordan algebra case. It relies on the construction of Lie triple systems associated with a Jordan one. Remark that using the notion of T ∗ −extension, we can construct solvable pseudoEuclidean Jordan (resp. quadratic Lie) triple systems as in Example 1. So, we can see that the set of pseudo-Euclidean Jordan (resp. quadratic Lie) triple systems contains strictly solvable and semi-simple Jordan (resp. Lie) triple systems. To relate quadratic Lie triple systems to pseudo-Euclidean Jordan triple systems, we shall recall in the second subsection of the third section the well-known TitsKantor-Koecher (TKK) construction, which relates Jordan systems to 3-graded Lie algebras. It is based on the observation that any Jordan triple system J , is associated to a Lie algebra Lie(J ) whose structure is closely connected with that of the initial Jordan triple system. This construction originated in the works of Tits [33], Kantor [22], Koecher [24] and Meyberg [30]. Further, it is well known that a Lie algebra (L, [ , ]) endowed with the triple product [a, b, c] := [[a, b], c], ∀a, b, c ∈ L is a Lie triple system. The converse is true (see [30]). Moreover, if we consider a quadratic Lie algebra (L, B) (that is a Lie algebra L with a symmetric nondegenerate invariant bilinear form B), then (L, [[ , ], ], B) is a quadratic Lie triple system. Conversely, a quadratic Lie triple system can be considered as a sub-Lie algebra of a quadratic Lie algebra [34].

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67

The purposes of this subsection is to prove that if the Jordan triple system is pseudo-Euclidean, then the Lie algebra constructed form it by TKK construction is quadratic. Then, using a result of [34] we prove that the Lie triple system arising from a pseudo-Euclidean Jordan triple systems by the TTK construction, is also quadratic. The fourth section is devoted to make an inductive description of quadratic Lie triple systems using the double extension. To generalize this notion to Lie triple systems, we need to introduce many new definitions as Definitions 6, 7 and 9. Moreover, many new propositions and lemmas are given. In this section, we investigate how we can extend this notion to triple systems. We seek to understand the structure of triple system viewing as a double extension and then demonstrate that a non- simple quadratic Lie triple system is a double extension of another one. After that, we prove that such a triple system can be reconstructed by double extension and/or direct sum of Lie triple systems of dimension one. To summarise, we shall prove in the fourth section the following results: Let U be the set constituted by {0} and the one-dimensional Lie triple system and let E be the set containing U and all simple Lie triple systems. • If (L, B) is a solvable quadratic Lie triple system, then L is obtained from elements L1 , . . . , Ln of E, by a finite number of orthogonal direct sums of quadratic Lie triple systems or/and double extensions by one-dimensional Lie triple systems. • If (L, B) is a quadratic Lie triple system, then L is obtained from elements L1 , . . . , Ln of E, by a finite number of orthogonal direct sums of quadratic Lie triple systems or/and double extensions by a simple Lie triple system or/and double extensions by one-dimensional Lie triple systems.

2 Preambles The results of this section hold for Lie triple systems and also for Jordan triple systems. We often delete the adjectives Jordan and Lie. We start by recalling some basic definitions. All definitions of this section are previous in the field of triple systems. A triple system as it is considered in this paper is a finite-dimensional linear space V together with a trilinear map (·, ·, ·) : V × V × V → V . A bilinear form B of a triple system V is said to be invariant or associative if B((x, y, z), u) = B(x, (u, z, y)) = B(z, (y, x, u)), ∀x, y, z, u ∈ V. A triple system is said to be a Jordan triple system if the trilinear map satisfies the following identities: {x, y, z} = {z, y, x}, {x, y, {u, v, w}} − {u, v, {x, y, w}} = {{x, y, u}, v, w} − {u, {y, x, v}, w}. The theory of Jordan triple systems is developed in [18, 20, 27, 28, 31]. A Lie triple system is a triple system with a trilinear map satisfying the following identities:

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[x, x, z] = 0, [x, y, z] + [y, z, x] + [z, x, y] = 0, [u, v, [x, y, z]] = [[u, v, x], y, z] + [x, [u, v, y], z] + [x, y, [u, v, z]]. For a more general class of triple systems, see [19, 30]. In the situation considered here, every Jordan (resp. Lie) triple system contains a unique maximal nilpotent ideal which is called the radical. A Jordan (resp. Lie) triple system is called semi-simple, if its radical vanishes. The fact that a Jordan (resp. Lie) triple system is semi-simple is equivalent to the fact that its trace form σ defined by σ(x, y) =

  1 Trace L(x, y) + L(y, x) , 2

(1)

(where L(x, y)(z) := {x, y, z}) is nondegenerate (see [18] and [32]). It is well known that σ is symmetric and invariant. A bilinear form on a Jordan (resp. Lie) triple system which is symmetric invariant and nondegenerate is called with misuse of language, invariant scalar product. A Jordan (resp. Lie) triple system endowed with such a bilinear form is said to be pseudo-Euclidean (resp. quadratic) see [4]. We often delete the adjective pseudoEuclidean (resp. quadratic). So that, a Jordan (resp. Lie) triple system implies in general pseudo-Euclidean (resp. quadratic) Jordan (resp. Lie) triple system). Definition 1 Let V be a triple system. 1. For a, b ∈ V, we define the linear maps L(a, b) (resp. R(a, b)) on V by L(a, b)x = (a, b, x) (resp. R(a, b)x = (x, a, b)), ∀x ∈ V. The linear maps L(a, b) (resp. R(a, b)) are called the left (resp. right) multiplications of V. 2. An endomorphism D of V is said to be a derivation if D((a, b, c)) = (D(a), b, c) + (a, D(b), c) + (a, b, D(c)), ∀a, b, c ∈ V. The set of all derivation of V is denoted by Der(V). The following proposition is due to Didry in [16]. Proposition 1 Let V be a triple system and W be a vector space. Let r, l : V × V −→ End (V) be two bilinear maps. The pair (r, l) is called a repW is endowed with the triple resentation of V in W if the Linear space V1 = V product: (a ⊕ x, b ⊕ y, c ⊕ z) = (a, b, c) ⊕ l(a, b)z + r(b, c)x − r(a, c)y,

(2)

∀a, b, c ∈ V, x, y, z ∈ W, is a triple system. In this case W is called a V−module. It was proved in [34] that if B is a symmetric and right-invariant bilinear form on V, (that is B(R(x, y)z, u) = B(z, R(y, x)u) for x, y, z ∈ V), then B is left invariant (that is B(L(x, y)z, u) = B(z, L(y, x)u) for x, y, z ∈ V). Therefore, a symmetric bilinear form B is invariant if and only if it is right invariant.

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Definition 2 1. Let (V, B) be a triple system. An endomorphism f of V is called B−symmetric, (resp. B−anti-symmetric) if B(f (a), b) = B(a, f (b)), (resp. B(f (a), b) = −B(a, f (b))), ∀a, b ∈ V. Denote by Ends (V) (resp. Enda (V)) the subspace of B−symmetric (resp. B−antisymmetric) endomorphism of V. 2. Let (V, B) be a triple system and W a linear space. A representation (r, l) of V in W is said to be B−anti-symmetric if B(l(x, y)a, b) = B(a, l(y, x)b) = −B(a, l(x, y)b), and B(r(x, y)a, b) = B(a, r(y, x)b), ∀a, b ∈ L, x, y ∈ W. Let us recall the following proposition from [16]. Proposition 2 Let V be a triple system, and W, W  be two vector spaces. 1. If R is the right multiplication of V and V is the left multiplication of V, then the pair (R, L) is a representation of V into itself called the regular representation of V. 2. Let (R, L) be the regular representation of V. The pair (R∗ , L∗ ) ∈ (End (V × V, End (V ∗ )))2 defined by: R∗ (x, y)(f ) = f ◦ R(y, x), L∗ (x, y)(f ) = f ◦ L(y, x), ∀x, y ∈ V, f ∈ L∗ , is a representation of V called the coregular representation of V. Rep(V, W  ). 3. Let (r1 , l1 ) ∈ Rep(V, W) and (r2 , l2 ) ∈  The pair (r, l) ∈ (End (V × V, End (W W  )))2 defined for all a, b ∈ V, v ∈  W, w ∈ W by: r(a, b)(v + w) = r1 (a, b)(v) + r2 (a, b)(w), l(a, b)(v + w) = l1 (a, b)(v) + l2 (a, b)(w),

is a representation of V in W



W .

Definition 3 Let V be a triple system and W, W  be two vector spaces. Two representations (r1 , l1 ) ∈ Rep(V, W) and (r2 , l2 ) ∈ Rep(V, W  ) are called equivalent if there exist an isomorphism of vector spaces φ : W −→ W  such that φ ◦ r1 (a, b) = r2 (a, b) ◦ φ and φ ◦ l1 (a, b) = l2 (a, b) ◦ φ, ∀a, b ∈ V. Definition 4 Let (V, ( , )) be a triple system. We define the descending series (V n )n∈N by V 0 = V and V n+1 = (V n , V n , V , ), ∀n ∈ N and the ascending series (V (n) )n∈N by V (0) = J and V (n+1) = (V (n) , V (n) , V (n) ), ∀n ∈ N. If there exists n ∈ N such that V n = {0} (resp. V (n) = {0}), then J is called solvable (resp. nilpotent).

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Definition 5 Let V be a triple system and B an invariant scalar product on V. 1. An ideal U of V is a subspace of V which satisfies (U, V, V) + (V, U, V) ⊆ U. 2. An ideal U of V is said to be a. Abelian if {U, U, U} = {0}. b. Solvable (resp. nipotent) if it is solvable (resp. nilpotent) as a Jordan triple system. c. Nondegenerate (resp. degenerate) if B|U ×U is nondegenerate (resp. degenerate). 3. The largest solvable ideal of V is called the radical of V and denoted as Rad (V). 4. The triple system (V, B) is called a. Semi-simple if it has no non-trivial solvable ideal. That is, Rad (V) = {0}. b. B-irreducible, if V contains no non-trivial nondegenerate ideal. Now, we shall prove some new results on the structure of triple systems: Proposition 3 Let (V, B) be a triple system and Z(V) = {a ∈ V; (a, b, c) = 0, ∀b, c ∈ V}  ⊥ be the centre of V. Then, Z(V) = (V, V, V), where (V, V, V) is the subspace of V spanned by the set {(a, b, c); a, b, c ∈ V}. Proof Let a, b, c ∈ V and let x ∈ Z(V). Then, B((a, b, c), x) = B(a, (x, c, b)) = 0.  ⊥ So, (V, V, V) ⊆ Z(V) . Conversely, let y ∈ (V, V, V)⊥ . Using the invariance of B we get, B((y, b, a), c) = 0, forc ∈ V. Thus, (y, b, a) = 0, for a, b ∈ V because B  ⊥ is nondegenerate. Hence, y ∈ Z(V). Consequently, Z(V) = (V, V, V).  The following lemma is straightforward. Lemma 1 Let (V, B) be a triple system and U be an ideal of V. Then, (i) U ⊥ = {x ∈ J , B(x, y) = 0 ∀y ∈ J } is an ideal of V. (ii) If U is nondegenerate, then J = U U ⊥ and U ⊥ is also nondegenerate. Lemma 2 Let (V, B) be a triple system. Then, V = and such that for i ∈ {1, . . . , r},

r  i=1

(i) Vi is a nondegenerate ideal of V. (ii) Vi is B−irreducible as a triple system. (ii) For i = j and (x, y) ∈ Vi × Vj , we have B(x, y) = 0.

Vi where r ∈ N

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Proof We precede by induction on n = dim(V). If n = 1, then the assertion is true. Suppose that every triple system of dimension less than n satisfies the proposition. Let (V, B) be a triple system of dimension n + 1. If V does not contain any non-trivial nondegenerate ideal, then the assertion is true for r =  1. If not, let I be a non-trivial nondegenerate ideal of V. By the Lemma 1, V = I I ⊥ . The result follows by ⊥  applying the induction to I and I . Proposition 4 Let (V, B) be a semi-simple triple system and consider the decomr  position V = Vi of V as in Lemma 2. i=1

(i) If I is a simple ideal of V, then there exists i0 ∈ {1, . . . , r} such that Vi0 = I. (ii) For any i ∈ {1, . . . , r}, Vi is simple. Proof (i) Let I be a non-trivial simple ideal of V. Assume that for all i ∈ {1, . . . , r}. r  We have I ∩ Vi = {0}. Since {I, V, V} ⊆ (I ∩ Vi ) = 0. Then, {I, I, I} = 0 and i=1

I is solvable. Hence, there exists i0 ∈ {1, . . . , r} such that I ∩ Vi0 = {0}. Since I ∩ Vi0 is an ideal of I and I is simple, then I ∩ Vi0 = I. So, I ⊆ Vi0 . The facts that Vi0 is B−irreducible and I is nondegenerate imply that I = Vi0 . (ii) Suppose that there exists i ∈ {1, . . . , r} such that Vi is not simple. Then, without s r   lost of generality, we may write V = ( Vi ) ⊕ ( Vi ) where for 1 ≤ i ≤ s, Vi is i=1

i=s+1

simple and Vi is not simple for s + 1 ≤ i ≤ r. Since V is semi-simple, then we can l  consider the decomposition V = Vi of V into the direct sum of its simple ideals. i=1

The assertion (i) implies that s = l = r.



The previous proposition shows that in the case of semi simple triple systems, the decomposition into the direct sum of orthogonal nondegenerate ideals coincides with the decomposition into a direct sum of simple ideals.

3 Pseudo-Euclidean Jordan Triple Systems 3.1 T∗ −Extension of Jordan Triple Systems The following theorem presents a process of construction of pseudo-Euclidean Jordan triple systems. ∗ Theorem 1 Let (J , { , , }J ) be a Jordan triple system and∗ w : J × J × J −→ J ∗ be a trilinear map. The vector space Tw (J ) = J J endowed with the triple product is defined by

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{x + f , y + g, z + h} = {x, y, z}J + w(x, y, z) + {f , y, z} + {x, g, z} + {x, y, h}, for all x, y, z ∈ J and f , g, h ∈ J ∗ . Where

(3)

{f , y, z}(a) = f ({a, z, y}); {x, g, z}(a) = g({x, a, z}) and {x, y, h}(a) = h({y, x, a}), is a Jordan triple system if and only if, ω satisfies w(z, y, x) = w(x, y, z), and w(x, y, {z, t, r}) − w(z, t, {x, y, r}) − w({x, y, z}, t, r) + w(z, {y, x, t}, r) = {w(x, y, z), t, r} − {x, y, w(z, t, r)} + {z, t, w(x, y, r)} − {z, w(y, x, t), r}, ∀x, y, z, t, r ∈ J . Furthermore, the bilinear form B defined on Tw∗ (J ) by B(x + f , y + g) = g(x) + f (y), ∀x, y ∈ J , f , g ∈ J ∗ is an invariant scalar product on Tw∗ (J ) if and only if w satisfies w(a, b, x)(y) = w(b, a, y)(x), ∀a, b, x, y ∈ J . The constructed pseudo-Euclidean Jordan triple system (Tw∗ (J ), B) is called the Tw∗ -extension of J by means of w. Proof Let x, y, u, v, a ∈ J and f ∈ J ∗ .     f , u, {x, y, v} − x, y, {f , u, v} (a)     − {f , u, x}, y, v + x, {u, f , y}v (a)     = f a, {x, y, v}, u − {y, x, a}, v, u     −f {a, v, y}, x, u − y, {x, a, v}, u      = f − y, x, {a, v, u} + a, v, {y, x, u}     −f {a, v, y}, x, u − y, {x, a, v}, u     = f {a, v, y}, x, u − y, {v, a, x}, u     −f {a, v, y}, x, u − y, {x, a, v}, u = 0.

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So { , , }|J ∗ ×J ×J , { , , }|J ×J ×J ∗ and { , , }|J ×J ∗ ×J satisfies the Jordan triple system conditions. Consequently, Tω∗ (J ) endowed with the product (3) is a Jordan triple system if and only if ω satisfies the given conditions. Besides, it is easy to verify that the bilinear form B is symmetric and nondegenerate on J . Moreover, B is invariant if and only if w(a, b, x)(y) = w(b, a, y)(x), ∀a, b, x, y ∈ J .  Remark 1 It is clear that J ∗ is an abelian ideal of T ∗ w J . Thus, T ∗ w J is not semisimple. Moreover, if J is not nilpotent, then T ∗ w J is not nilpotent too. Consequently, the family of pseudo-Euclidean Jordan triple systems contains strictly the families of semi-simple Jordan triple systems and pseudo-Euclidean nilpotent Jordan triple systems. Theorem 2 Let (J , B) be a pseudo-Euclidean Jordan triple system of dimension n. Then, (J , B) is isometric to a T ∗ -extension (Tw∗ (J1 ), B1 ) if and only if n is even and J contains an isotropic ideal I of dimension 2n . Proof Let I be an isotropic ideal of J of dimension 2n . Since B is nondegenerate, the ideal Let us consider V an isotropic complementary to I. Then, J =  I is abelian. I V and V ⊥ = V. Let x, y, z ∈ V. Put {x, y, z} = α(x, y, z) + β(x, y, z), where α(x, y, z) ∈ I and β(x, y, z) ∈ V. It is easy to check that (V, β) is a Jordan triple system. Now, since B is nondegenerate, the linear map ν : I −→ V ∗ ; i −→ B(i, .) is invertible. Furthermore, dim(I) = 2n = dim(V ∗ ). Thus, ν is an isomorphism of vector spaces. On the other hand, the trilinear map w : V × V × V −→ V ∗ defined by w(x, y, z) = ν(α(x, y, z)), x, y, z ∈ V is a 3−cocycle of V. In fact, w(x, y, z)(v) = ν(α(x, y, z))(v) = B(α(x, y, z), v) = B({x, y, z}, v), ∀x, y, z, v ∈ V .

 ∗ ∗ ∗ Thus, we can V of V by means of w and consider the T ∗-extension Tw (V) = V :J =I V −→ V V ; i + x −→ x + ν(i). Using the invariance of the bilinear form B on J , we get for all x, y, z ∈ V, i ∈ I,         ν(i) {x, y, z} = B i, {x, y, z} = B {i, z, y}, x = ν {i, z, y} (x). Similarly,

      ν(i) {x, y, z} = ν {z, i, x} (y) = ν {y, x, i} (z).

  ∗ x −→ x + ν(i), ∀i ∈ Consequently, if  : J = I V −→ V V defined by i +  I, x ∈ V then, for X = i + x, Y = j + y, Z = k + z ∈ J = I V, we get

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   {X , Y , Z}

        = β(x, y, z) ⊕ ν α(x, y, z) + ν {x, y, k} + ν {i, y, z} + ν {x, j, z}       = β(x, y, z) ⊕ w(x, y, z) + ν(k) {y, x, .} + ν(i) {., z, y} + ν(j) {z, ., x}   = (X ), (Y ), (Z) .

Thus,  is an isomorphism of Jordan triple systems.



Theorem 3 Let (J , B) be a pseudo-Euclidean Jordan triple system of dimension n. If n is odd, and I is an isotropic ideal of J of dimension [ 2n ]. Then, J is isomorphic to a nondegenerate ideal of codimension 1 in a T ∗ -extension of the Jordan triple system J /I. Proof Let I be an isotropic ideal of J of dimension [ 2n ] and let L(J ) be the Lie triple system generated by the left, right and middle multiplications of J . Since B is invariant on J . By Lemma 4.2 of [25], φ(I ⊥ ) ⊆ I, for all φ ∈ L(J ). Consequently, {J , J , I ⊥ } + {J , I ⊥ , J } + {I ⊥ , J , J } ⊆ I. So,

 ⊥ I ⊥ ⊆ {J , J , I ⊥ } + {J , I ⊥ , J } + {I ⊥ , J , J } .

Since B is invariant and nondegenerate, then {J , J , I ⊥ } + {J , I ⊥ , J } + {I ⊥ , J , J } = {0}. Therefore, I ⊥ is abelian. Now, let us consider the one-dimensional abelian Jordan triple system Kc endowed with the bilinear form Bc : Kc × Kc −→ K defined by Bc (c, c) = 1. Let J1 = J ⊕ Kc be the Jordan triple system endowed with the triple product given by {x + αc, y + γc, z + λc} = {x, y, z}, ∀x, y, z ∈ J , α, γ, λ ∈ K. We define on J1 the bilinear form B1 by B1|J ×J = B,

B1 (c, c) = 1, and

B1 (J , c) = B1 (c, J ) = 0.

It is clear that (J1 , B1 ) is a pseudo-Euclidean Jordan triple system. Besides, J1 is a ⊥ nondegenerate ideal of codimension 1 of J1 . Let d ∈ I such that B(d , d ) = −1, Ke, where e = c + d . It is easy to see that I1 is an ideal of and consider I1 = I J1 . In addition, B1 (e, e) = B(d , d ) + Bc (c, c) = 0, and B1 (x, e) = B(x, d ) + B(x, c) = 0, ∀x ∈ I.

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So, I1 is isotropic and, dim(I1 ) = n+1 . Since dimension of J1 is even, then Theorem 2 2 implies that J1 is isomorphic to a T ∗ -extension of the Jordan triple system J1 /I1 .  Consequently, J1 /I1 is isomorphic to J /I.

3.2 Tits-Kantor-Koecher of Pseudo-Euclidean Jordan Triple Systems A quadratic Lie algebra is a Lie algebra with a nondegenerate, symmetric and invariant bilinear form. It is known that any Jordan triple system is associated to a quadratic Lie algebra of larger dimension called the Lie algebra obtained by the Tits-KantorKoecher construction (TKK construction) (see [30]). In the following theorem, we recall the result mentioned in [30]. Theorem 4 Let J be a Jordan triplesystem and consider the subspace h  of  End (J ) End (J ) spanned by the set: l(x, y) := L(x, y) ⊕ −L(y, x), x, y ∈ J . For a, b ∈ J , wehave l(x, y)(a ⊕ b) = L(x, y)a ⊕ −L(y, x)b. The vector space Lie(J ) := J h J (where J is a copy of J ) endowed with the bracket: [x ⊕ y, a ⊕ b] = l(x, b) − l(a, y), [l(x, y), l(a, b)] = [L(x, y), L(a, b)] ⊕ [L(y, x), L(b, a)], [l(a, b), x ⊕ y] = L(a, b)x ⊕ −L(b, a)y, ∀x, y, a, , b, ∈ J , is a Lie algebra called the TKK algebra of J . Now, we prove that if J is pseudo-Euclidean, then Lie(J ) is quadratic. First, by an easy calculation, we can prove the following lemma: Lemma 3 Let (J , B) be a pseudo-Euclidean Jordan triple system. Then, (i) [h, l(x, y)] = l(hx, y) + l(x, hy), ∀x, y ∈ J , ∀h ∈ h. (ii) B(l(x, y)u, v) = B(l(u, v)x, y), ∀x, y, u, v ∈ J . Theorem 5 Let (J , B) be a pseudo-Euclidean Jordan triple system. On the TKK algebra Lie(J ) = J ⊕ h ⊕ J of J , we define the symmetric bilinear form BL by: For x, y, u, v ∈ J , BL (x, y) = B(x, y), BL (l(x, y), l(u, v)) = B(l(x, y)u, v), BL (h, J ⊕ J ) = {0}, BL (J , J ) = BL (J , J ) = {0}. The bilinear form BL is invariant, nondegenerate and symmetric on Lie(J ) or (Lie(J ), BL ) is a quadratic Lie algebra. Moreover, BL is the unique invariant nondegenerate symmetric bilinear form on Lie(J ) satisfying

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BL (J , J ) = BL (J , J ) = BL (h, J



J ) = 0 and BL (x, y) = B(x, y), ∀x, y ∈ J .

(4) Proof Using (i) of Lemma 3, we can show by an easy calculation that BL is well defined and symmetric. Now, let h1 , h2 ∈ h and u, v ∈ J . Then, BL ([h1 , l(u, v)], h2 ) − BL (h1 , [l(u, v), h2 ]) = 2(BL (h2 h1 u, v) + BL (h2 u, h1 v)). The fact that all the elements of h are anti-symmetric with respect to B implies that BL ([h1 , l(u, v)], h2 ) = BL (h1 , [l(u, v), h2 ]). By an easy computation, we obtain that BL is invariant. Since B is nondegenerate, then BL is also nondegenerate. Suppose that there exists another  on Lie(J ) satisfying (4). Then, for all x, y, ∈ J , h ∈ h, invariant scalar product BL we have    (l(x, y), h) = BL ([x, y], h) = BL (x, [y, h]) BL  = −BL (x, hy) = B(hx, y) = BL (l(x, y), h).   = BL|(h×h) . Consequently, BL = BL . So, BL| (h×h)



Lemma 4 Let J be a Jordan triple system over an algebraically closed field K and Lie(J ) be its TKK algebra. If Lie(J ) is simple, then J is simple. Proof Since Lie(J ) is simple, then J is semi-simple (Theorem 7 of [30]). So, {J , J , J } = 0 and P(J ) = {0}. Now, let B and B be two elements of P(J ). Consider the invariant scalar products φ and φ on Lie(J ) associated, respectively, to B and B as in Theorem 5. That is,  (x, y) = B (x, y), ∀x ∈ J , ∀y ∈ J . BL (x, y) = B(x, y) and BL

Since Lie(J ) is simple, then by the Corollary 3.1 of [3], there exists α ∈ K such  = αBL . that BL  (x, y) = αBL (x, y) = αB(x, y). So, Therefore, for all x, y ∈ J , B (x, y) = BL Theorem 1.3 of [8] implies that J is a simple Jordan triple system.  Theorem 6 Let J be a unitary Jordan triple system and Lie(J ) be its TKK algebra. Then, J is simple if and only if Lie(J ) is a simple Lie algebra. Proof If Lie(J ) is simple, then by Lemma 4, J is simple. Conversely, let BL and  be two nondegenerate invariant symmetric bilinear forms on Lie(J ) and define BL the two bilinear forms B and B on J by: B(x, y) = BL (x, y), ∀x, y ∈ J ,  (x, y), ∀x, y ∈ J . B (x, y) = BL

It is clear that B and B’ are invariant. Besides, the fact that J is unital implies that B and B are symmetric on J . Since J is simple, then by Theorem 1.3 of [8], every

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symmetric invariant bilinear form on J is nondegenerate. Consequently, B and B are two invariant scalar products on J and there exists α ∈ K such that B = αB. Now, since J is a unitary Jordan triple system, we can easily show that:  − αBL )(J , J (BL



 h) = 0 and (BL − αBL )(J , J



h) = 0.

 − αBL = 0. Consequently, Lie(J ) has a unique quadratic structure. By Hence, BL Corollary 3.1 of [3], Lie(J ) is a simple Lie algebra. 

4 Quadratic Lie Triple Systems 4.1 Inductive Description of Quadratic Solvable Lie Triple Systems We start by the following proposition which gives a new characterisation of quadratic Lie triple systems. Proposition 5 Let L be a Lie triple system. Then, L is a quadratic Lie triple system if and only if the regular representation and the coregular representation are equivalent. Proof Let L be a Lie triple system, (R, L) and (R∗ , L∗ ) be the regular and the coregular representations of L. Suppose that L is a quadratic Lie triple system, then there exists a symmetric invariant nondegenerate bilinear form B : L × L −→ K. Therefore, the map φ : L −→ L∗ defined by: φ(a)(b) = B(a, b), ∀a, b ∈ L, is an isomorphism of vector spaces which satisfies (φ ◦ L(a, b)(c))(d ) = B(L(a, b)c, d ) = B(c, L(b, a)d ) = (φ(c) ◦ L(b, a))(d ) = L∗ (a, b)(φ(c))(d ), ∀a, b, c, d ∈ L. By the same way, we can show that (φ ◦ R(a, b)c)(d ) = R∗ (a, b)(φ(c))(d ). Therefore, (R, L) and (R∗ , L∗ ) are equivalent. Conversely, assume that (R, L) and (R∗ , L∗ ) are equivalent. Then, there exists an isomorphism of vector spaces φ : L −→ L∗ such that (φ ◦ R(a, b)c)(d ) = R∗ (a, b)(φ(c))(d ) and (φ ◦ L(a, b)c)(d ) = L∗ (a, b)(φ(c))(d ), ∀a, b, c, d ∈ L. Let us define the bilinear map T : L × L −→ K by T (a, b) = φ(a)(b), ∀a, b ∈ L.

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Since φ is invertible, then T is nondegenerate. Besides, T is left invariant. In fact, for all a, b, c, d ∈ L, T (L(a, b)c, d ) = φ(L(a, b)c)(d ) = L∗ (a, b)(φ(c))(d ) = φ(c)(L(b, a)(d )) = T (c, L(b, a)d ). Similarly, we can verify that T is right invariant. So, T is invariant on L. But, T is not necessarily symmetric. Consider the symmetric (resp. anti-symmetric) part Ts (resp. Ta ) of T defined by: Ts (x, y) =

1 1 (T (x, y) + T (y, x)), Ta (x, y) = (T (x, y) − T (y, x)), ∀x, y ∈ L. 2 2

It is clear that T is invariant if and only if Ts and Ta are invariant. Let us consider Ls = {x ∈ L; Ts (x, y) = 0, ∀y ∈ L} and La = {x ∈ L; Ta (x, y) = 0, ∀y ∈ L}. The fact that T is invariant and nondegenerate implies that Ls ∩ La = {0} and Ls and La are ideals of L. Let a, b, c, d ∈ L. Since Ta is invariant, then Ta ([a, b, c], d ) = −Ta (d , [a, b, c]) = −Ta ([d , c, b], a) = Ta ([c, d , b], a) = Ta (c, [a, b, d ]) = −Ta (c, [b, a, d ]) = −Ta ([a, b, c], d ). Thus, Ta ([a, b, c], d ) = 0. It follows that L3 := [L, L, L] is contained in La . Consequently, L3s = [Ls , Ls , Ls ] ⊆ Ls ∩ La = {0}.  Ls . ConNow, let U be a sub-vector space of L such that La ⊆ U and L = U sider a symmetric nondegenerate bilinear form F on Ls . Since L3s = [Ls , Ls , Ls ] = {0}, then F is invariant. Therefore, the bilinear form B : L × L −→ K is defined by B|U ×U = Ts|U ×U , B|Ls ×Ls = F,

B(U, Ls ) = B(Ls , U) = {0},

is symmetric invariant and nondegenerate on L. So, (L, B) is a quadratic Lie triple system.  Now we shall give an example of quadratic Lie triple system using the T∗ − extension of Lie triple systems. Example 1 Let L be a two-dimensional vector space and {e1 , e2 } be a basis of L. Consider the triple product on L defined by [e1 , e2 , e1 ] = −[e2 , e1 , e1 ] = e2 (all other brackets are assumed to be null). Then, (L, [ , , ]) is a Lie triple system. Let w : L × L × L −→ L∗ be the trilinear map defined by w(e1 , e1 , e1 ) = e1∗ . We verify that w is an alternating 3-cocycle.Then, we can consider the four-dimensional quadratic Lie triple system Tw∗ (L) = L L∗ , T ∗ -extension of L by mean of w. Since L is solvable non-nilpotent, then so is Tw∗ (L) [25].

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It is not hard, by some easy computation, to prove the following theorem: Theorem 7 Let (L, [ , , ]) be a Lie triple system, V be a  vector space and φ :  = L V is endowed with L × L × L −→ V be a 3−cocycle. The vector space L the following triple product: [a ⊕ x, b ⊕ y, c ⊕ z] = [a, b, c] ⊕ φ(a, b, c), ∀a, b, c ∈ L, x, y, z ∈ V,

(5)

is a Lie triple system called the central extension of L by V (by means of φ). Lemma 5 Let (L, B) be a quadratic Lie triple system and F : L −→ Enda (L); u −→ F(u, .) be a linear map satisfying     B a, G(b, c) = −B b, F(a, c) ,

(6)

where G(a, b) = F(b, a) − F(a, b), ∀a, b, c ∈ L. The trilinear map φ : L × L × L −→ K defined by   φ(a, b, c) = B a, F(c, b) , ∀a, b, c ∈ L is a 3−cocycle if and only if the following condition is verified: [a, b, F(c, u)] = F([a, b, c], u) + [G(a, b), c, u] + F(c, [a, b, u]), ∀a, b, c, u ∈ L. Proof Let a, b, c, u, v ∈ L. It is easy to see that the map G(a, b) is anti-symmetric. Therefore, φ(a, b, c) + φ(b, a, c) = B(b, F(c, a)) + B(a, F(c, b)) = −B(c, G(b, a) + G(a, b)) = 0.

In addition, φ(a, b, c) + φ(b, c, a) + φ(c, a, b) = B(a, F(c, b)) + B(b, F(a, c)) + B(c, F(b, a)) = B(a, F(c, b) − G(b, c)) − B(b, G(c, a)) = B(a, F(b, c)) − B(b, G(a, c)) = −B(b, G(a, c) + G(c, a)) = 0.

Consequently, the map φ satisfies (Co1) and (Co2). Let us verify Co3: φ([a, b, c], u, v) + φ(c, [a, b, u], v) + φ(c, u, [a, b, v]) − φ(a, b, [c, u, v]) = B(c, −[a, b, F(v, u)] + F(v, [a, b, u]) + F([a, b, v], u)) + B([c, u, v], G((a, b)) = B(c, −[a, b, F(v, u)] + F(v, [a, b, u]) + F([a, b, v], u) + [G(a, b), v, u]).



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Definition 6 Let (L, B) be a quadratic Lie triple system. The linear map F : L −→ Enda (L); a −→ F(a, .) is called an admissible function on L if it satisfies the following conditions: F(a, .) ∈ Der(L) and (E1 ) [a, b, F(c, u)] = F([a, b, c], u) + [c, G(b, a), u] + F(c, [a, b, u]), ∀a, b, c, u ∈ L.

If moreover there exists D ∈ End (L) such that the following conditions are satisfied: (E2 ) [a, b, D(u)] = D([a, b, u]) + G(u, G(a, b)) + F(u, G(b, a)), (E3 ) F(a, D(u)) = D(F(a, u)) − G(u, D(a)) + F(u, D(a)), (E4 ) F(a, F(b, c)) = F(F(a, b), c) + F(b, F(a, c)) − [D(a), b, c], ∀a, b, c, u ∈ L,

where G(a, b) = F(b, a) − F(a, b), then (D, F) is called a generalised representation of Kd in L. Remark 2 Let (L, B) be a quadratic Lie triple system, Kd be a one-dimensional Lie triple system and F be an admissible function on L satisfying Eq. (6). Then, the  trilinear map  φ : L × L × L −→ Kd defined for all a, b, c ∈ L by φ(a, b, c) = = B a, F(c, b) d is a 3−cocycle. So, we can consider the Lie triple system L  ∗ L Kd , central extension of L by Kd by means of φ. It is not hard, by some computation, to prove the following proposition: Proposition 6 Let (L, B) be a quadratic Lie triple system, Kd be a one- dimensional Lie triple system  and (D, F) be a generalised representation in L. Then, the vector  = L Kd endowed with the product defined for all a, b, c ∈ L, α, β, γ ∈ space L K by [a + αd , b + βd , c + γd ] = [a, b, c] + αF(b, c) − βF(a, c) + γG(a, b) − αγD(b) + βγD(a)

(7) is a Lie triple system called the generalised semi-direct product of L by the onedimensional Lie triple system Kd . Let us recall the following proposition of [16]. Proposition 7 Let L be a Lie triple system and V be a vector space. Let r, l be two linear maps from L × L into End (V ). The pair (r, l) is a representation of L in V if and only if it satisfies

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(R1) l(x, y) = −l(y, x), (R2) l(x, y) = r(y, x) − r(x, y), (R3) l(x, y)l(u, v) − l(u, v)l(x, y) = l([x, y, u], v) + l(u, [x, y, v]), (R4) l(x, y)r(u, v) − r(u, v)l(x, y) = r([x, y, u], v) + r(u, [x, y, v]), (R5) r(x, [z, u, v]) = r(u, v)r(x, z) − r(z, v)r(x, u) + l(z, u)r(x, v), for allx, y, z, u, v ∈ L. The set of all representations of L in V will be denoted by Rep(L, V ). Remark 3 Let L be a Lie triple system and V be a vector space. Let r, l be two linear maps from L × L into End (V ). If r, l verify (R2 ), (R4 ) and (R5 ), then (r, l) is a representation of L in V. Theorem 8 Let (L, B) be a quadratic Lie triple system, Kd be a one- dimensional Lie triple system and (D, F) be a generalised representation in L which satisfies 1. D is a symmetric endomorphism of L, 2. F(a, derivation of L, ∀a ∈ L,  .) is an anti-symmetric    3. B a, G(b, c) = −B b, F(a, c) , ∀a, b, c ∈ L.   (i) The vector space L = Kd L Kd ∗ endowed with the following triple product: [αd + a + α d ∗ , βd + b + β  d ∗ , γd + c + γ  d ∗ ] = [a, b, c] + αF(b, c) − βF(a, c) + γG(a, b) − αγD(b) + βγD(a) +αB(D(b), c)d ∗ − βB(D(a), c)d ∗ + B(a, F(c, b))d ∗ , for all a, b, c ∈ L, α, β, γ, α , β  , γ  ∈ K is a Lie triple system. (ii) The bilinear form B : L × L −→ K defined by B|L×L = B, B(d , d ∗ ) = 1, B(d , d ) = B(d ∗ , d ∗ ) = 0, B(d , L) = B(d ∗ , L) = {0}, is an invariant scalar product on L.     The quadratic Lie triple system L = Kd L Kd ∗ , B is called the double extension of L by Kd by means of (D, F). Proof (i) Since the admissible function F satisfies the condition (6). By Remark 3,   = L Kd ∗ endowed with the triple product defined by the vector space L [a + αd ∗ , b + γd ∗ , c + γd ∗ ] = [a, b, c] + B(a, F(c, b)), ∀a, b, c ∈ L, α, β, γ ∈ K,

 −→ End (L)  by is a Lie triple system. Let us define the linear map  F :L  F(a + αd ∗ )(b + βd ∗ ) = F(a, b) + B(D(a), b), ∀a, b ∈ L, α, β ∈ K.

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The fact that (D, F) satisfies the condition (E4 ) imply that  F(a + αd ∗ ) belongs  for all a ∈ L, α ∈ K. Using the conditions (E2 ) and (6) we get: For all to Der(L) a, b, c, u ∈ L         B a, F(F(c, u), b) = B D([a, b, c]), u + B c, F(u, G(b, a)) + B D(c), [a, b, u] .

So,  F satisfies the condition (E1 ) because F satisfies the condition (E1 ). Consequently,  F is an admissible function of Kd in L. Let us define the endomorphism  D  by on L  D(a) = D(a) and  D(d ∗ ) = 0, ∀a ∈ L. D,  F) satisfies the conSince (D, F) satisfies the condition (E2 ) (resp. (E3 )), then ( dition (E2 ) (resp. (E3 )). In addition, using the conditions (E3 ) and (6) we prove that   ( D,  F) satisfies   the condition (E4 ). Therefore, (D, F) is a generalised representation of Kd on L Kd ∗ . So, we can consider the semi-direct product of L Kd ∗ by Kd by means of ( D,  F). (ii) It is easy to see that the bilinear form B is symmetric and nondegenerate. Further, the fact that D is a symmetric endomorphism of L and F(a, .) is an antisymmetric derivation of L for all a ∈ L implies that B is invariant. To conclude, B  is an invariant scalar product on L.  Theorem 9 Let (L, B) be a quadratic irreducible Lie triple system. If Z(L) = {0}, then L is a double extension of a quadratic Lie triple system by a one-dimensional Lie triple system. Proof Let e ∈ Z(L)\{0} and put I = Ke. Since L is irreducible and I is a minimal ideal, then I ⊥ is a maximal ideal of L and I ⊆ I ⊥ . So, B(e, e) = 0. The fact that B is nondegenerate that B(d , e) =1, B(d , d ) =  implies that there exists d∈ L such ⊥ ⊥ Kd . Consider W = (Kd Ke) . Then, I⊥ = W Ke and L = 0 and L = I   Kd W Ke. Let x, y, z ∈ W, then [x, y, z] = β(x, y, z) + α(x, y, z)e where β(x, y, z) ∈ W and α(x, y, z) ∈ K. It is easy to chek that (W, β) is a Lie triple system and B|W×W is nondegenerate. Thus, (W, β, B|W×W ) is a quadratic Lie triple system and the trilinear map α : W × W × W −→ K is a 3−cocycle. Now, let a, b ∈ W. Then, [a, d , d ] = D(a) + ϕ(a)e and [d , a, b] = F(a, b) + ψ(a, b)e where D ∈ End (W), F : W −→ End (W) and ϕ : W −→ K is a linear form. The fact that L verifies the third condition of Lie triple systems implies that F(a, .) ∈ Der(L), ∀a ∈ L and the pair (D, F) satisfies the conditions (E2 ), (E3 ) and (E4 ). Besides, since α is a 3−cocycle, then (D, F) satisfies the conditions (E1 ) and (6). Thus, the pair (D, F) is a generalised representation of Kd on L. Moreover, using the invariance of the bilinear form B on L we get D is a symmetric endomorphism of L and F(a, .) is an anti-symmetric derivation of L for all a ∈ L. Consequently,   the Lie triple system Kd W Kd ∗ is the double extension of (W, B|W×W ) by

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the one-dimensional Lie triple system Kd by means of (D, F). By the invariance of B, we get φ(a) = 0, ψ(a, b) = B(D(a), b) and α(a, b, c) = B(a, F(c, b)), ∀a, b, c ∈ L.   Which implies that L is isomorphic to the double extension Kd W Kd ∗ of (W, B|W×W ) by the one-dimensional Lie triple system Kd by means of (D, F).  Corollary 1 Let (L, B) be a solvable irreducible quadratic Lie triple system. Then, L is a double extension of a solvable quadratic Lie triple system by a one- dimensional Lie triple system. Proof Since L is solvable, then [L, L, L] = L. So, Z(L) = {0}. By the preceding Theorem, L is a double extension of a quadratic Lie triple system W = I ⊥ /I by a one-dimensional Lie triple system, where I = Kd ⊆ Z(L).  Let U be the set constituted by {0} and the one-dimensional Lie triple system. Theorem 10 Let (L, B) be a solvable quadratic Lie triple system. If L ∈ / U, then L is obtained from elements L1 , . . . , Ln of E, by a finite number of orthogonal direct sums of quadratic Lie triple systems or/and double extensions by a one-dimensional Lie triple systems. Proof We proceed by induction on dim(L). If dim(L) = 0 or 1, then L ∈ U. Suppose that dim(L) = 2. Then L is either the orthogonal direct sum of two one-dimensional Lie triple systems or L is the double extension of {0} by an one-dimensional Lie triple system. Now, assume that the theorem is satisfied for dim(L) < n ∈ N. We shall prove it in the case where dim(L) = n. If L is irreducible, then L is a double extension of a solvable Lie triple system (W, T ) by an one-dimensional Lie triple system. The fact that dim(W) < n, implies that (W, T ) satisfies the theorem. If L is not irreducible, then L = L1 . . . Ln where Li , i ∈ {1, . . . , n} are non-zero nondegenerate irreducible ideals of L such that B(Li , Lj ) = {0}, ∀i = j ∈ {1, . . . n}. Since dim(Li ) < dim(L),  then (Li , B|Li ×Li ) satisfies the theorem.

4.2 Inductive Description of Quadratic Lie Triple Systems Lemma 6 Let (L, B) be a quadratic Lie triple system and V be a Lie triple system. Let π1 : V × L × L −→ L be a trilinear map such that π1 (v, a, .) ∈ Enda (L), ∀v ∈ V, a ∈ L, B(a, π1 (u, c, b)) = B(c, π2 (b, a, u)), ∀a, b, c ∈ L, u ∈ V,

(8)

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where π2 : L × L × V −→ L is the map defined by π2 (a, b, v) = π1 (v, b, a) − π1 (v, a, b), ∀a, b ∈ L, ∈ V . The map φ : L × L × L −→ V ∗ defined by φ(a, b, c)(v) = B(a, π1 (v, c, b)), ∀a, b, c ∈ L, v ∈ V, is a 3−cocycle if and only if π1 and π2 satisfy: for a, b, c, d ∈ L and v ∈ V : [a, b, π1 (v, c, d )] = π1 (v, [a, b, c], d ) + [c, π2 (b, a, v), d ] + π1 (v, c, [a, b, d ]). = L In this case, the vector space L



V ∗ endowed with the product:

[a ⊕ f , b ⊕ g, c ⊕ h] = [a, b, c] ⊕ φ(a, b, c), ∀a, b, c ∈ L, is a Lie triple system central extension of L by V. Proof Let a, b, c, d , e ∈ L and v ∈ V. Then, φ(b, a, c)(v) = B(b, π1 (v, c, a)) = −B(a, π1 (v, c, b)) = −φ(a, b, c)(v). φ satisfies the condition (Co1).

Thus,

  φ(a, b, c) + φ(b, c, a) + φ(c, a, b) (v) = B(a, π1 (v, c, b)) + B(b, π1 (v, a, c)) + B(c, π1 (v, b, a)) = B(a, π1 (v, c, b)) + B(b, π1 (v, a, c)) − B(a, π1 (v, b, c)) = B(a, π2 (b, c, v)) + B(a, π2 (c, b, v)) = 0 So, φ satisfies the condition (Co2). Furthermore, 

 φ([a, b, c], d , e) + φ(c, [a, b, d ], e) + φ(c, d , [a, b, e]) (v)   = B c, −[a, b, π1 (v, e, d )] + π1 (v, e, [a, b, d ]) + π1 (v, [a, b, e])

= −B(c, [e, π2 (b, a, v), d ]) = B(π2 (a, b, v), [d , c, e]) = φ(a, b, [c, d , e]).  Definition 7 Let L and V be two Lie triple systems. Let π1 : V × L × L −→ L be a trilinear map and π2 : L × L × V −→ L be the map defined by π2 (a, b, v) = π1 (v, b, a) − π1 (v, a, b), ∀a, b ∈ L, ∈ V . The map π1 is called an admissible action of V on L if it satisfies

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1. π1 (x, a, .) is a derivation of L, for all x ∈ V , a ∈ L. 2. π2 (a, b, .) is a 1 − cocycle. 3. π1 (v, [a, b, c], d ) + π1 (v, c, [a, b, d ]) = [a, b, π1 (v, c, d )] − [c, π2 (b, a, v), d ], ∀a, b, c, d ∈ L, ∈ V . Definition 8 Let L and V be two Lie triple systems and (r, l) be a representation of V in L. Given an admissible action π1 : V × L × L −→ L of V on L. We say that π1 is compatible with the representation (r, l) if the following compatibility’s conditions are satisfied: (C1 ) [a, b, r(u, v)c] = r(u, v)[a, b, c] + π2 (c, π2 (a, b, u), v) + π1 (u, c, π2 (b, a, u)), (C2 ) l(v, w)π1 (u, a, b) = π1 ([v, w, u], a, b) + π1 (u, l(v, w)a, b) + π1 (u, a, l(v, w)b), (C3 ) π1 (u, a, r(v, w)b) = r(v, w)π1 (u, a, b) − π2 (b, r(u, v)a, w) + π1 (v, b, r(u, w)a, (C4 ) π1 (u, a, π1 (v, b, c)) = [b, r(u, v)a, c] + π1 (v, π1 (u, a, b), c) + π1 (v, b, π1 (u, a, c)),

∀a, b, c ∈ L, u, v, w ∈ V. Lemma 7 Let (L, B) be a quadratic Lie triple system, V be a Lie triple system and (r, l) be an anti-symmetric representation of V in L such that l(a, b) is a derivation of L for all a, b in L. Let π1 : V × L × L −→ L be an admissible action of V on L compatible with the representation (r, l) satisfying the condition B(a, π1 (u, c, b)) =  B(c, π2∗(b, a, u)), ∗∀a, b, c ∈ L, u ∈ V. Let us consider the central extension L = L V of L by V . If π1 (v, a, .) is a B-anti-symmetric endomorphism of L for all v in × L  −→ L  defined by  π1 (v, a + f , b + g) = V and a in L, then the map  π1 : V × L ∗ π1 (v, a, b) + B(a, r(., v)b), ∀a, b ∈ L, f , g ∈ V , v ∈ V is an admissible action of  V on L. Proof Thanks to the skew symmetry of the derivation π1 (u, a, .) and the condition (C4 ) of compatibility we may write   φ(π1 (u, a, b), c, d ) + φ(b, π1 (u, a, c), d ) + φ(b, c, π1 (u, a, d )) (v)   = B b, −π1 (u, a, π1 (v, d , c)) + π1 (v, d , π1 (u, a, c) + π1 (v, π1 (u, a, d ), c     = −B b, [d , r(u, v)a, c] = B a, r(v, u)[b, c, d ] , ∀a, b, c, d ∈ L, u, v ∈ V.  Using the Since π1 (u, a, .) ∈ Der(L), ∀u ∈ V, a ∈ L, then π1 (v, a + f , .) ∈ Der(L). Eq. (8) and the condition (C1 ) of compatibility, we get

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φ(π2 (a, b, u), c, d )(v) − φ(a, b, π1 (u, c, d ))(v) = B(d , π2 (c, π2 (a, b, u), v) − B(π1 (u, c, d ), π2 (b, a, v)) = B(d , π2 (c, π2 (a, b, u), v) − π1 (u, c, π2 (b, a, v)) = r(u, v)[a, b, c] + [b, a, r(u, v)c]). × L × π2 : L Thus,  π1 satisfies the condition (8). Further, it is clear that that the map   defined by V −→ L  π2 (a + f , b + g, v) =  π1 (v, a + f , b + g) −  π1 (v, b + g, a + f ), ∀a, b ∈ L, f , g ∈ L∗ , v ∈ V

satisfies:  π2 (a + f , b + g, v) = π2 (a, b, v) + B(a, l(., v)b), ∀a, b ∈ L, f , g ∈ L∗ , v ∈ V. So, it becomes easy to see that  π2 is a 1−cocycle if and only if, for all a, b ∈ L, and z, u, v, x ∈ V we have B(a, l(x, [z, u, v])b) = B(a, l([x, v, u], z)b) + B(a, l([v, x, z], u)b) + B(a, l([u, z, x], v)b). The fact that l(x, y) satisfies the equations (R3 ) and (R1 ) implies that       B a, l(x, [z, u, v])b − B a, l([u, z, x], v)b = B a, (l(x, v)l(u, z) − l(u, z)l(x, v))b     = B a, l([x, v, u], z)b + B a, l([v, x, z], u)b .

 By an easy computation, we can prove the following proposition: Proposition 8 Let L and V be two Lie triple systems and (r, l) be a representation of V in L such that l(x, y) belong to Der(L) for all x, y ∈ V. Let π 1 : V × L × L −→ L  = L V is endowed with be an admissible action of V on L. The vector space L the triple product defined for all a, b, c ∈ L, u, v, w ∈ V by [a ⊕ u, b ⊕ v, c ⊕ w] = [u, v, w] ⊕ l(u, v)c + r(v, w)a − r(u, w)b +π2 (a, b, w) + π1 (u, b, c) − π1 (v, a, c) + [a, b, c], is a Lie triple system if and only if π1 is compatible with the representation (r, l).  is called the semi-direct product Definition 9 In this case, The Lie triple system L of L by V by means of (r, π1 ) and (r, π1 ) is said to be a generalised representation of V in L.

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Remark 4 1. The triple product [ , , ] : L × L × L −→ L of a Lie triple system (L, [ , , ]) is an admissible action compatible with the regular representation (R, L) of L. Then, the pair (R, [ , , ]) is a generalised representation of L in itself. 2. We say that the generalised representation (r, π1 ) is anti-symmetric if (r, l) is a B-anti-symmetric representation and π1 (x, a, .) is an anti-symmetric derivation of L for all x ∈ V, a ∈ L. Theorem 11 Let L and V be two Lie triple systems, B be an invariant scalar product on L and γ be a symmetric invariant bilinear form on V. Let (r, π1 ) be an antisymmetric generalised representation of V in L. Let us put φ(a, b, c)(v) = B(a, π1 (v, c, b)) and π2 (a, b, v) = π1 (v, b, a) − π1 (v, a, b), for a, b, c ∈ L, v ∈ V. Suppose that π1 satisfies the condition: B(a, π1 (u, c, b)) = B(c, π2 (b, a, u)), ∀a, b, c ∈ L, u ∈ V. Then,  = V ⊕ L ⊕ V ∗ endowed with the triple product defined for (i) The vector space L u, v, z ∈ V, a, b, c ∈ L, f , g, h ∈ V ∗ by [u + a + f , v + b + g, z + c + h] = [u, v, z] ⊕ [a, b, c] + r(v, z)a − r(u, z)b + l(u, v)c +π2 (a, b, z) − π1 (v, a, c) + π1 (u, b, c) ⊕φ(a, b, c) + f ◦ R(z, v) + h ◦ L(v, u) − g ◦ R(z, u) +B(a, l(., z)b) − B(a, r(., v)c) + B(b, r(., u)c), is a Lie triple system. (ii) The the bilinear form  B : L1 × L1 −→ K, defined for x, y ∈ V, a, b ∈ L, f , g ∈ V ∗ by  B(x + a + f , y + b + g) = γ(x, y) + B(a, b) + f (y) + g(x),  is an invariant scalar product on L.  Proof (i) Lemma 6 implies that we can consider the Lie triple system L1 = L V ∗  is defined central extension of L by V by means of φ. Recall that the product on L by [a ⊕ f , b ⊕ g, c ⊕ h] = [a, b, c] ⊕ φ(a, b, c), ∀a, b, c ∈ L.  be the bilinear maps defined for all u, v ∈ V, a, b ∈ L, Let r, l : V × V −→ End (L) ∗ f , g ∈ V by

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 r(u, v)(a + f ) = r(u, v) + f ◦ R(v, u),  l(u, v)(a + f ) = l(u, v) + f ◦ L(v, u).  By Proposition 2, ( r, l) is a representation of V in L. Since π1 satisfies the condition (C2 ), then φ(a, b, c) ◦ L(v, u) = φ(l(u, v)a, b, c) + φ(a, l(u, v)b, c) + φ(a, b, l(u, v)c), ∀a, b, c ∈ L, u, v ∈ V. So, the fact that l(u, v) belongs to Der(L) for all u, v ∈ V  for all u, v ∈ V. Now, let us define the map  π1 : implies that l(u, v) belongs to Der(L)    V × L × L −→ L by π1 (v, a + f , b + g) = π1 (v, a, b) + B(a, r(., v)b), ∀a, b ∈ L, f , g ∈ V ∗ , v ∈ V. Lemma 7 implies that π1 is an admissible action of V on L. r, l) veriMoreover, π1 satisfies the condition (C1 )and π2 is a 1-cocycle. Thus, (  r, l) also satisfies fies (C1 ). Furthermore, (r, π1 ) satisfies (C2 ) (resp.(C3 )). Thus, ( (C2 ) (resp.(C3 )) because (r, l) verifies (R4 ) (resp.(R5 )). Finally, (C4 ) is verified due π1 is compatible with the to the equalities (C2 ) and (8). So, the admissible action   Conserepresentation ( r, l) and ( r, π1 ) is a generalised representation of V on L. ∗ quently, we can consider the Lie triple system V ⊕ L ⊕ V the semi-direct product r, π1 ). of L ⊕ V ∗ by V by means of ( (ii) It is clear that the bilinear form  B is symmetric. Besides, the skew symmeB is try of the generalised representation (r, π1 ) and the condition (8) imply that  right invariant. Moreover, since B is nondegenerate then it is easy to check that  B is nondegenerate.   Definition 10 The quadratic Lie triple system (L, B) constructed in Theorem 11 is called the double extension of L by V by means of (r, π1 ).  Theorem 12 Let (L, B) be an irreducible quadratic Lie triple system. If L = I V where I is a maximal ideal of L and V is a subsystem of L. Then, L is the double extension of the quadratic Lie triple system (W = I/I ⊥ ,  B) where  B(X , Y ) = B(X , Y ), ∀X , Y ∈ L. Proof The ideal I is maximal, so I ⊥ is a minimal ideal of L. Since L is irreducible, ⊥ ⊥ V. It follows that I = I ⊆ I. Consider A = I then I ∩ I ⊥ = {0}. Therefore,   I ⊥ A⊥ and L = I ⊥ A⊥ V. Now, let a, b, c ∈ A⊥ , then [a, b, c] = α(a, b, c) + β(a, b, c) where α(a, b, c) ∈ ⊥ I and β(a, b, c) ∈ A⊥ . It easy to chek that A⊥ endowed with the trilinear map β is a Lie triple system. Moreover, (A⊥ , Q = B|A⊥ ×A⊥ ) is a quadratic Lie triple system. The map θ = s|A⊥ : A⊥ −→ I/I ⊥ is an isomorphism of Lie triple systems. Let a ∈ A⊥ , u, v ∈ V. Then, there is i ∈ I ⊥ and c ∈ A⊥ such that [a, u, v] = i + c. Thus, ∀x ∈ V B(i, x) = B([a, u, v] − c, x) = B([a, u, v], x) = B(a, [x, v, u]) = 0.

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So, i = 0 and [A⊥ , V, V] ⊆ A⊥ . Similarly, we show that [V, V, A⊥ ] ⊆ A⊥ . Consequently, the maps R1 , L1 : V × V −→ End (A⊥ ) defined by R1 (u, v)a = R(u, v)a and L1 (u, v)a = L(u, v)a are well defined and the pair (R1 , L1 ) belong to Rep(V, A⊥ ). Now, let us consider the map π1 : V × A⊥ × A⊥ −→ A⊥ defined by π1 [u, a, b] = (θ−1 ◦ s)([u, a, b]), ∀a, b ∈ A⊥ , u ∈ V. By Remark 4, π1 is an admissible action of V on A⊥ which is compatible  withthe representation (R1 , L1 ). Hence, we can consider the double extension V ∗ A⊥ V of A⊥ by V by means of (R1 , π1 ). Define the map ν : I ⊥ −→ V ∗ (resp. δ : V −→ (I ⊥ )∗ ) by ν(i) = B(i, .), ∀i ∈ I ⊥ (resp. δ(v) = B(v, .), ∀v ∈ V). Since B is nondegenerate, then ν (resp.δ) is one to ∗ = dimV isomorphism of vector spaces. one. Hence, dimI ⊥   and ν is an   ⊥ ⊥ A V −→ V ∗ A⊥ V; (i + a + v) −→ ν(i) + a + v The map ∇ : I is an isomorphism of Lie triple system. In fact, let a = i + u, Y = b + j + v, and Z = c + l + w be three elements X = of L = I ⊥ A⊥ V, where a, b, c ∈ A⊥ , i, j, l ∈ I ⊥ , u, v, w ∈ V. Then, [X , Y , Z] = [u, v, w] + β(a, b, c) + π2 (a, b, w) − π1 (v, a, c) + π1 (u, b, c) +R1 (v, w)a − R1 u, w)b + L(u, v)c + α(a, b, c) + [i, v, w] +[u, j, w] + [u, v, l] + α1 (a, b, w) + α2 (a, v, c) + α3 (u, b, c), where π2 (a, b, w) = π1 (w, b, a) − π1 (w, a, b), α1 (a, b, w) = [a, b, w] − π2 (a, b, w), α2 (a, v, c) = [a, v, c] + π1 (v, a, c), α3 (u, b, c) = [u, b, c] − π1 (u, b, c). It is easy to chek that ν([i, v, w]) = ν(i) ◦ R(w, v) and ν([u, v, l]) = ν(l) ◦ L(v, u). So, ∇([X , Y , Z]) = [u, v, w] + β(a, b, c) + π2 (a, b, w) − π1 (v, a, c) +π1 (u, b, c) + R1 (v, w)a − R1 u, w)b + L(u, v)c +ν(α(a, b, c)) + ν(i) ◦ R(w, v) − ν(j) ◦ R(w, u) + ν(l) ◦ L(v, u) +ν(α1 (a, b, w)) + ν(α2 (a, v, c)) + ν(α3 (u, b, c)). Besides, for x ∈ V ν(α(a, b, c)(x) = B([a, b, c] − β(a, b, c), x) = B(a, α3 (u, b, c) + π1 (u, b, c)) = B(a, π1 (x, c, b)).

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Define the trilinear map φ : A⊥ × A⊥ × A⊥ −→ V ∗ by φ(a, b, c) −→ B(a, π1 (., c, b)) ∀a, b, c ∈ A⊥ . The following assertions are straightforward: ν(α1 (a, b, w)) = B(a, L(., w)b), ν(α2 (a, v, c)) = −B(a, R(., v)c), ν(α3 (u, b, c)) = B(b, R(., u)c). Therefore, ∇([X , Y , Z]) = [∇(X ), ∇(Y ), ∇(Z)]. Hence, ∇ is an isomorphism of Lie triple systems.  Corollary 2 Let (L, B) be a irreducible quadratic Lie triple system. If L is not solvable nor simple, then L is a double extension of a quadratic Lie triple system (W, T ) by a simple Lie triple system.  Proof Since L is not solvable, then L = S Rad (L), where Rad (L) is the radical n  Si is a semi-simple subsystem of L. of L and S = i=1    So, I = S2 . . . Sn Rad (L) is a maximal ideal of L. Theorem 12 imply that  L is the double extension of W = I /I ⊥ by S1 . Corollary 3 Let (L, B) be a quadratic irreducible non-simple Lie triple system. If Z(L) = {0}, then L is a double extension of a quadratic Lie triple system (W, T ) by a simple Lie triple system. Proof If Z(L) = {0}, then Z(L)⊥ = L. So, [L, L, L] = L. Therefore, L is not solvable. By the preceding Corollary, L is a double extension of a quadratic Lie triple system (W, T ) by a simple Lie triple system.  Let E be the set constituted by {0}, the one-dimensional Lie triple system and all simple Lie triple systems. Theorem 13 Let (L, B) a quadratic Lie triple system. If L ∈ / E, then L is obtained from elements L1 , . . . , Ln of E, by a finite number of orthogonal direct sums of quadratic Lie triple systems or/and double extensions by a simple Lie triple system or/and double extensions by one-dimensional Lie triple systems. Proof We proceed by induction on dim(L). If dim(L) = 0 or 1, then L ∈ E. Assume that dim(L) = 2. If L is not irreducible, then L = L1 L2 where L1 , L2 are two nondegenerate ideals of L which satisfies B(L1 , L2 ) = {0} and dim(L1 ) = dim(L2 ) = 1. Therefore, L1 , (resp. L2 ) is the one-dimensional Lie triple system. Now, suppose that L is irreducible. Then, L is a double extension of {0} by an one-dimensional Lie triple system. We conclude that if dim(L) = 2, the theorem is satisfied.

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Now, we suppose that the theorem is satisfied for dim(L) < n ∈ N. We shall prove it in the case where dim(L) = n. If L ∈ / E and L is irreducible, then L is a double extension of a quadratic Lie triple system (W, T ) by either a simple Lie triple system or the one-dimensional Lie triple system. Since dim(W) < dim(L), then (W, T ) satisfies the theorem and so(L, B) satisfies it. If L is not irreducible, then L = L1 . . . Ln where Li , i ∈ {1, . . . , n} are nonzero nondegenerate irreducible ideals of L such that: B(Li , Lj ) = {0}, ∀ i = j ∈ {1, . . . n}. Since dim(Li ) < dim(L), then (Li , B|Li ×Li ) satisfies the theorem. Thus, (L, B) satisfies the theorem. 

5 Conclusion The results of this chapter give a new process to construct all quadratic Lie triple systems starting from the Lie triple systems of dimension 1 using just the direct summation and the double extension. In the future, we can use this inductive description to construct lower dimension triple systems and generalise the work of T. B. Bouetou in [15]. This work will be a subject of joint work with H. Bouraoui. Moreover, we have proved that every Jordan triple system of even dimension, which contains an isotropic ideal of dimension 2n , is a T ∗ -extension of a Jordan triple system and every odd-dimensional Jordan triple system is an ideal of codimension one of a T ∗ -extension. This construction can be extended to symplectic Lie or Jordan algebras. We have also related quadratic Lie triple systems to pseudo-Euclidean Jordan triple systems by proving the well-known Tits-Kantor-Koecher construction which relates Jordan systems to 3-graded Lie algebras. We have proved that if the Jordan triple system is pseudo-Euclidean, then the Lie algebras constructed form it by the Tits-Kantor-Koecher construction is quadratic. Then, using a result of [34] we have proved that the Lie triple system arising from a pseudo-Euclidean triple systems by the Tits-Kantor-Koecher construction is also quadratic. Acknowledgements The authors would like to thank Deanship of Scientific Research at Umm Al-Qura University (Project ID 18-SCI-1-01-0023) for financial support. We would like to thank also Saïd Benayadi for useful discussions; the editor and the referee for their valuable comments which helped to improve the manuscript.

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References 1. Alvarez, M.A., Rodriguez-Vallarte, M.C., Salgado, G.: Low dimensional contact Lie algebras. J. Lie Theory 29(3), 811–838 (2019) 2. Ait Ben Haddou, M., Boulmane, S.: Pseudo-Euclidean alternative algebras. Commun. Algebra 44(12), 5199–5222 (2016) 3. Bajo, I., Benayadi, S.: Lie algebras admitting a unique quadratic structure. Commun. Algebra 25(9), 2795–2805 (1997) 4. Baklouti, A.: Quadratic Hom-Lie triple systems. J. Geom. Phys. 121, 166–175 (2017) 5. Baklouti, A., Benayadi, S.: Symmetric symplectic commutative associative algebras and related Lie algebras. Algebra Colloq. 18(spec. 1), 973–983 (2011) 6. Baklouti, A., Benayadi, S.: Pseudo-euclidean Jordan algebras. Commun. Algebra 43, 2094– 2123 (2015) 7. Baklouti, A., Benayadi, S.: Symplectic Jacobi Jordan algebra. Linear Multilinear Algebra (2019). https://doi.org/10.1080/03081087.2019.1626334 8. Baklouti, A., Hidri, S.: Semi-simple Jordan and Lie triple systems. Preprint 9. Baklouti, A., Ben Salah, W., Mansour, S.: Jordan superalgebras and associative scalar product. Commun. Algebra. 41, 2441–2466 (2013) 10. Beites, P.D., Kaygorodov, I., Popov, Y.: Generalized derivations of multiplicative n-Ary Hom-ω color algebras. Bull. Malays. Math. Sci. Soc. 42(1), 315–335 (2019) 11. Bertram, W.: The Geometry of Jordan and Lie Structures. Lecture Notes in Mathematics, vol. 1754. Springer (2000) 12. Benayadi, S., Bouarroudj, S.: Double extensions of Lie superalgebras in characteristic 2 with nondegenerate invariant supersymmetric bilinear form. J. Algebra 510, 141–179 (2018) 13. Bordemann, M.: Nondegenerate invariant bilinear forms on nonassociative algebras. Acta Math. Univ. Comenian. 2, 151–201 (1997) 14. Bouarroudj, S., Leites, D., Shang, J.: Computer-aided study of double extensions of restricted Lie superalgebras preserving the nondegenerate closed 2-forms in characteristic 2. Experiment. Math. (2019). https://doi.org/10.1080/10586458.2019.1683102 15. Bouetou, T.B.: Classification of solvable 3-dimensional Lie triple systems. In: Sabinin, L., Sbitneva, L., Shestakov, I. (eds.) Non-Associative Algebra and Its Applications. Chapman and Hall/CRC, New York (2006). https://doi.org/10.1201/9781420003451 16. Didry, M.: Structures algebriques associees aux espaces symetriques. Thesis. http://www.iecn. u-nancy.fr/edidrym/ (2006) 17. Hopkins, N.C.: Some structure theory for a class of triple systems. Trans. Amer. Math. Soc. 289, 203–212 (1985) 18. Jacobson, N.: Lie and Jordan triple systems. Amer. J. Math. 71, 149–170 (1949) 19. Kamiya, N.: Lie triple system. Encyclopaedia Math. Sci. (2001) 20. Kamiya, N.: Jordan triple system. Encyclopaedia Math. Sci. (2001) 21. Kamiya, N., Okubo, S.: On triple systems and Yang-Baxter equations. Proc. Seventh Inter. Colloq. Diff. 189–196 (1997) 22. Kantor, I.L.: Transitive differential groups and invariant connections on homogenous spaces. Trudy Sem. Vecktor. Tenzor. Anal. 13, 310–398 (1966) 23. Kaygorodov, I., Popov, Y.: Split regular hom-leibniz color 3-algebras. Colloq. Math. 157(2), 251–277 (2019) 24. Koecher, M.: Imbedding of Jordan algebras into Lie algebras. I. Amer. J. Math. 89, 787–816 (1967) 25. Lin, L., Wang, Y., Deng, S.: T*-extension of Lie triple systems (2009). https://doi.org/10.1016/ j.laa.2009.07.001 26. Lister, W.G.: A structure theory of Lie triple systems. Trans. Amer. Math. Soc. 72, 217–242 (1952) 27. Loos, O.: Lectures on Jordan triples. Lec. Notes. Univ. British Comb. Vancouver (1971) 28. Loos, O.: Jordan pairs. Lec. Notes Math. (1975)

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Comparative Study of Some Numerical Methods for the Standard FitzHugh-Nagumo Equation ˙ Koffi Messan Agbavon, Appanah Rao Appadu, and Bilge Inan

Abstract We use a few methods to solve the standard FitzHugh–Nagumo equation with specified initial and boundary conditions. The methods used are three versions of nonstandard finite difference (NSFD) schemes and two finite difference schemes constructed from the exact solution. This work is an improvement and extension of the work in Namjoo and Zibaei (Comput Appl Math 1–17, [1]). We would like to point out that Namjoo and Zibaei derived two schemes using the exact solution but the schemes were not tested on numerical experiments. We study some properties of the five methods such as stability, positive definiteness and boundedness. The performance of five methods is compared by computing L 1 , L ∞ errors and rate of convergence for two values of γ (which controls the global dynamics of the equation (Xu et al. in 2014 36th Annual International Conference of the IEEE, Engineering in Medicine and Biology Society (EMBC), pp. 4334–4337 [2])), namely 0.001, 0.5 for small and large spatial domains at time T = 1.0.

K. M. Agbavon Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 0002, South Africa e-mail: [email protected] A. R. Appadu (B) Department of Mathematics and Applied Mathematics, Nelson Mandela University, University Way, Summerstrand, Port Elizabeth 6031, South Africa e-mail: [email protected] B. ˙Inan Department of Mathematics Education, Muallim Rıfat Faculty of Education, Kilis 7 Aralık University, 79000 Kilis, Turkey e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 D. Zeidan et al. (eds.), Computational Mathematics and Applications, Forum for Interdisciplinary Mathematics, https://doi.org/10.1007/978-981-15-8498-5_5

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1 Introduction Partial differential equations are widely used to describe or model the complex phenomena in real life and applications are in fluid mechanics, solid-state physics, plasma wave and chemical physics [3, 4]. The standard FitzHugh–Nagumo equation is an important application of nonlinear partial differential equation and used to model the transmission of nerve impulses [5, 6]. It is also used in circuit theory, biology and population genetics [7] as mathematical models. It is given by u t − u x x = u(1 − u)(u − γ),

(1)

where γ controls the global dynamics of the equation and is in the interval (0, 1) [2] and u(x, t) is the unknown function which depends on the temporal variable, t and the spatial variable, x.

Background The first fruitful mathematical representation of electrophysiology was the main model established by Hodgkin and Huxley in the early 1950s. Hodgkin and Huxley carried out voltage-clamp experiments on a squid giant axon and changed their studies of transmembrane potential, currents and conductance into a circuit-like model. The result was a system of four ordinary differential equations (ODEs) that precisely portrayed noticeable propagation alongside an axon. Although the complex Hodgkin–Huxley equations have shown to be a good model to describe a signal propagation along a nerve, they are hard to be analysed. FitzHugh [5] and Nagumo [6] tackled this problem a decade later when they restrained the original system of four variables down to a simpler model of only two variables. Their simple model is much easier to be analysed and it still describes the main phenomena of the dynamics: (1) a sufficiently large stimulus will start off a significant response, and (2) after such a stimulus and response, the medium needs a period of recuperation time before it can be impulsed again. These two properties are characterised as excitable (γ < 0 : the nerve is in excitable mode) and refractory (γ > 0 : the nerve is in refractory mode and does not respond to external stimulation). Excitation happens fast, while recovery occurs slowly. A medium that displays excitability and refractoriness is categorised as an excitable medium. Though the standard FitzHugh–Nagumo equation is one of the simplest models, it shows complex dynamics that have not been fully investigated. For example, it endorses a stable travelling and pulse solution. However, the pulse can be destabilised by large perturbations. This, therefore, drew attention from many researchers. Moreover, the study showed that when γ ∈ (0, 1) [8], the result would be heterozygote inferiority. When γ = −1, Eq. (1) is known as Newell–Whitehead equation and it describes dynamical behaviour near the bifurcation point for the Rayleigh–Benard

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convection of binary fluid mixtures. It has also been proved that the exact solution of Eq. (1) describes the fusion of two travelling fronts of the same sense and converts into a front which connects two stable constant states. Moreover, Jackson [9] used Galerkin’s approximations to solve Eq. (1). Bell et al. [10] studied the singular perturbation of N-front travelling waves while Gao et al. [11] studied the existence of wavefronts and impulses and Krupa et al. [12] studied fast and slow waves of Eq. (1). Furthermore, Shonbek [13] investigated the higher order derivatives of solutions of Eq. (1), whereas Chou et al. [14] studied the exotic dynamic behaviour of Eq. (1). Eq. (1) has been solved by many methods, namely Adomian Decomposition Method (ADM), Homotopy Perturbation Method (HPM), Variational Iteration Method (VIM), Homotopy Analysis Method (HAM), Differential Transform Method (DTM) and Nonstandard Finite Difference Method (NSFD). We will briefly describe the work performed using these methods in the following paragraph. ADM has been introduced by George Adomian in early 1980s [15] to show a new way to solve nonlinear functional equations. The method consists of separating the equation under study into linear and nonlinear portions which produce a solution in the form of a sequences. HPM is used to study the accurate asymptotic solutions of nonlinear problems [16–19]. VIM is used to identify the unknown function in parabolic equation [20–22]. DTM is a semi-analytical numerical method that makes use of Taylor series to obtain solutions of differential equations [23]. Nonstandard Finite Difference Scheme (NSFD) have been introduced by Mickens [24] to obtain solutions of various partial differential equations especially those in mathematical biology. The derivations are mostly based on the idea of dynamical consistency [25] (positivity, boundedness, monotonicity of the solutions, etc). After generalising these results, Mickens formulated the following three basic rules in constructing NSFD schemes: (1) The order of discrete derivatives should be equal to the order of corresponding derivatives appearing in the differential equation. (2) Discrete representation for derivatives, in general have non trivial denominator functions, for instance − u nj u n+1 j (2) ut ≈ φ(t, λ) where φ(t, λ) = t + O(t 2 ).

(3)

(3) Nonlinear terms must be represented by nonlocal discrete representations. For instance,   u j−1 + u j + u j−1 2 2 u j, (4) u j ≈ u j u j+1 , , u j ≈ 3 and u 3 ≈ 2u 3j − u 2j u j+1 , u 3j ≈ u j−1 u j u j+1 .

(5)

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2 Organisation of the Paper The paper is organised as follows. In Sect. 3, we describe the numerical experiment chosen. In Sect. 4, we provide some reasons why the construction of exact schemes is important. In Sect. 5, we give two schemes constructed by Namjoo and Zibaie using the exact scheme to obtain some properties of the scheme. We present some numerical results and tabulate some errors and this is a novelty in our work. In Sects. 6, we describe two nonstandard finite difference methods (NSFD1 and NSFD2) and obtain an improvement for NSFD1 which we call NSFD3 (second novelty in this work) and study some of the properties of the methods. We then present L 1 , L ∞ errors and results on the rate of convergence which is the third highlight of this work. Section 7 summarises the salient features of the paper. All simulations are performed by MATLAB 2014a software on an Intel core2 as CPU.

3 Numerical Experiment We solve Eq. (1), where γ ∈ (0, 1) and u(x, t) is the unknown function which depends on spatial variable, x ∈ [0, b] and temporal variable, t. The initial condition is u(x, 0) =

γ , 1 + e−2 A1 x

(6)

and the boundary conditions are given by γ , 1 + e 2 A1 A2 t γ , u(b, t) = −2 1 + e A1 (b−A2 t) u(0, t) =

where A1 =

√ 2 γ, 4

A2 =

4−2γ γ

(7)

A1 and t ∈ [0, 1]. The exact solution is given by [26]

u(x, t) =

γ 1+

e−2 A1 (x−A2 t)

.

(8)

In Namjoo and Zibaei [1], they used γ = 0.001, b = 1.0. We test the performance of the schemes over different values of γ and also over short and long domains at time, T = 1.0 We considered short and long domain as some methods might work for short domains and produce less efficient results for longer domain domains. We also consider a very small of γ as well as γ = 0.5. We consider 4 cases: Case 1 Case 2 Case 3 Case 4

:γ :γ :γ :γ

= 0.001, b = 1.0. = 0.001, b = 10.0. = 0.5, b = 1.0. = 0.5, b = 10.0.

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Definition 1(Miyata [27]) For a vector x ∈ R N , L 1 and L ∞ norms are defined by N |x i | and  x ∞ = max{|x i |, i = 1, · · ·N }.  x 1 = i=1 Definition 2 (Sutton [28]) Suppose {t n }0N forms a partition of [0, T ], with tn = nt for n = 0, · · ·N , where t = T /N . Suppose a vector x ∈ R N , defined by  1  x  L P (0,t n−1 ) +τ (x n ) p p for p ∈ [0, ∞),  x  L P (0,t n ) = max{ x  L P (0,t n−1 ) , x n } for p = ∞.

(9)

The rate of convergence with respect to time is defined by ratei (t) =

log(x i (t)) − log(x i−1 (t)) . log(t i ) − log(t i−1 )

4 Construction of Numerical Scheme from Exact Solution The equation u t + u x = 0 and u x = b u x x have known exact finite difference models which are u nj − u nj−1 − u nj u n+1 j + = 0, with k = h (10) k h and

u j − u j−1 u j+1 − u j + u j−1   =b , h b eh/b − 1 h

(11)

respectively, where h = x and k = t represent spatial mesh size and temporal step size, respectively. However, there no exact finite difference schemes for most ordinary or partial differential equations including the standard FitzHugh–Nagumo equation, Fisher’s and linear advection equations. In Appadu et al. [29], they solve the equation u t + u x = α u x x for 0 < x < 1, t > 0 with boundary condition u(0, t) = u(1, t) = 0 and initial condition u(x, 0) = 3 sin(4 π x). The numerical experiment is from Chawla et al. [30] and the exact solution is 1 t u(x, t) = e[ 2α (x− 2 )]

∞ 

ξj e



− α2j π 2 t

sin( j π x),

(12)

j=1

where

3  1 ξj = 1 + (−1) j+1 e−( 2 α )  2α

 1 2 2α

1 + ( j − 4)2 π 2

−

 1 2 2α

1 + ( j + 4)2 π 2

 .

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Three numerical methods were used in Appadu et al. [29], namely fourth-order upwind, NSFD and third-order upwind. They considered three different values of α, namely 0.01, 0.1 and 1.0 using h = 0.1 at some values of k. Dispersive oscillations were observed with all the three methods when α = 0.01. Quite good results were obtained when α = 0.1. and α = 1.0. This suggests that one cannot use a given numerical method to solve a certain partial differential equation for any value of the parameters controlling advection and diffusion in this case. It is for this reason that the authors believe that the approach used by Namjoo and Zibaei [1] to construct schemes from the exact solution should be explored. This approach is fairly new. Zhang et al. [31] constructed finite difference schemes for Burgers and Burgers–Fisher equations using the exact solution.

5 Scheme of Namjoo and Zibaei We describe how Namjoo and Zibaie constructed an explicit and implicit scheme from the work in [1]. The analytical solution is u(x, t) =

γ γ {1 + tanh[A1 (x − A2 t)]} = . 2 1 + e−2 A1 (x−A2 t)

(13)

and boundary and initial conditions can be obtained from Eq. (13) . We use the exact solution from Eq. (13) in order to obtain approximations for u x , u t and u x x . We make use of non-traditional denominators. The following forward difference approximations are used, namely ∂x u =

u(x + h, t) − u(x, t) u(x, t + k) − u(x, t) and ∂t u = , ψ1 φ2

and the backward difference approximations are used, namely u(x, t) − u(x − h, t) u(x, t) − u(x, t − k) and ∂¯t u = . ∂¯x u = ψ2 φ1 We note that ψ1 , φ2 , ψ2 and φ1 are non-traditional denominators to be determined. Using Eq. (13), we have u(x + h, t) =

γ 1+

e−2 A1 (x+h−A2 t)

, u(x, t − k) =

γ 1+

e−2 A1 (x−A2 t+A2 k)

.

(14)

we choose h = A2 k, we have u(x + h, t) = u(x, t − k) and u(x − h, t) = u(x, t + k). We have, therefore,

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1 1 + e−2 A1 (x−A2 t) 1 + e−2 A1 (x+h−A2 t) 1 − = − u(x, t) u(x + h, t) γ γ −2 A1 (x−A2 t)   e 1 − e−2 A1 h = γ     γ 1 = − 1 1 − e−2 A1 h γ u(x, t)   1 1 − (1 − e−2 A1 h ). = u(x, t) γ

(15)

We can now deduce that 1 1 − = u(x, t) u(x − h, t)



 1 1 − (1 − e2 A1 h ). u(x, t) γ

(16)

Since u(x, t + k) = u(x − h, t) and h = A2 k, using Eq. (16), we have 1 1 − = u(x, t) u(x, t + k)



 1 1 − (1 − e2 A1 A2 k ). u(x, t) γ

(17)

Also since u(x, t − k) = u(x + h, t) and h = A2 k, using Eq. (15), we have 1 1 − = u(x, t) u(x, t − k)



 1 1 − (1 − e−2 A1 A2 k ). u(x, t) γ

(18)

Using Eq. (15) , we have  u(x + h, t) − u(x, t) =

 1 1 − (1 − e−2 A1 h )u(x, t)u(x + t, t), (19) u(x, t) γ

which can be rewritten as   u(x, t) − (1 − e−2 A1 h )u(x + t, t). u(x + h, t) − u(x, t) = 1 − γ

(20)

We have therefore   u(x + h, t) − u(x, t) u(x, t) , = 2 A1 u(x + h, t) 1 − ∂x u = ψ1 γ where ψ1 =

1−e−2 A1 h 2 A1

≈ h.

Using Eqs. (16)–(18) and proceeding in the same manner, we obtain u¯ x , u t , u¯ t

(21)

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  u(x, t) − u(x − h, t) u(x, t) , = 2 A1 u(x − h, t) 1 − ∂¯x u = ψ2 γ   u(x, t + k) − u(x, t) u(x, t) −1 , ∂t u = = 2 A1 A2 u(x, t + k) φ2 γ   u(x, t) − u(x, t − k) u(x, t) −1 . ∂¯t u = = 2 A1 A2 u(x, t − k) φ1 γ

(22) (23) (24)

where ψ2 =

e 2 A1 h − 1 1 − e−2 A1 A2 k e 2 A1 A2 k − 1 ≈ h, φ1 = ≈ k, φ2 = ≈ k. 2 A1 2 A1 A2 2 A1 A2

(25)

We need to obtain an approximation for u x x .

5.1 Explicit Scheme We choose u x x = ∂¯x ∂x u and this combination generates an explicit scheme. Indeed, we have   u(x + h, t) − u(x, t) ∂¯x u(x + h, t) − ∂¯x u(x, t) ¯ ¯ u x x = ∂x ∂x u = ∂x = . (26) ψ1 ψ1 From Eqs. (21) and (22) , we can easily deduce that uxx =

2 A1 u(x, t) ψ1

    2 A1 u(x + h, t) u(x, t) − (27) 1− u(x − h, t) 1 − γ ψ1 γ

and it follows that 2 A1 2 A1 u(x, t) (u(x − h, t) − u(x + h, t)) . (u(x, t) − u(x − h, t)) + ψ1 γ ψ1 (28) We introduce A2 into Eq. (28) by adding and subtracting the expression

uxx =

A2 (u(x, t) − u(x − h, t)) , ψ1 so that u x x can be expressed in terms of a time derivative with other expressions. We then have

Comparative Study of Some Numerical Methods …

uxx =

103

2 A1 + A2 2 A1 u(x, t) (u(x − h, t) − u(x + h, t)) (u(x, t) − u(x − h, t)) + ψ1 γ ψ1 A2 − (u(x, t) − u(x − h, t)) . ψ1

(29) Since h = A2 k, we deduce that

A2 ψ1

=

1 . φ1

Also u(x, t + k) = u(x − h, t), therefore,

A2 u(x, t) − u(x, t + k) . (u(x, t) − u(x − h, t)) = ψ1 φ1

(30)

This gives uxx =

u(x, t + k) − u(x, t) 2 A1 + A2 + (u(x, t) − u(x − h, t)) φ1 ψ1 2 A1 u(x, t) (u(x − h, t) − u(x + h, t)) , + γ ψ1

(31)

which can be rewritten as uxx =

  u(x, t + k) − u(x, t) u(x, t) − u(x − h, t) + (2 A1 + A2 ) φ1 ψ1   u(x − h, t) − u(x, t) u(x, t) − u(x + h, t) 2 A1 . u(x, t) + + γ ψ1 ψ1

(32)

From Eqs. (21) ) and (22) , we have uxx =

  u(x, t + k) − u(x, t) u(x − h, t) + 2 A1 (2 A1 + A2 ) u(x, t) 1 − φ1 γ      u(x − h, t) u(x, t) 2 A1 u(x, t) −2 A1 u(x, t) 1 − − 2 A1 u(x + h, t) 1 − . + γ γ γ

(33) We remark that since A1 = 





√

2 γ 4

and A2 =



4−2γ γ



A1 , we have



γ 1 γ2 , 2 A1 (2 A1 + A2 ) = + γ (4 − 2γ) = γ. 2 2 4 (34) Therefore, by using (34), (33) becomes 2 A1

uxx =

2 A1 γ

=

4 γ

2 2 γ 16

=

u(x, t + k) − u(x, t) u 2 (x, t) + (u(x + h, t) + u(x − h, t)) φ1 2 γ − u(x, t) (u(x, t) + u(x + h, t)) + γ u(x, t) − u(x, t) u(x − h, t). (35) 2

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We can approximate u x x by a second-order central difference approximation using a non-traditional denominator and thus obtain an explicit scheme which discretises Eq. (1) . Using Eq. (35) , we have u nj+1 − 2u nj + u nj−1 ψ1 ψ2

=

− u nj u n+1 j φ1

 + (u nj )2

u nj+1 + u nj−1 2



 − (γ u nj ) −

u nj + u nj+1

u nj u nj−1



2 + γ u nj , (36)

which can be rewritten as  φ1 (u nj )2  n  φ1  n u j+1 + u nj−1 u j+1 − 2u nj + u nj−1 − ψ1 ψ2 2 u nj  n  u j + u nj+1 + φ1 u nj u nj−1 − φ1 γ u nj . + φ1 γ 2

u n+1 = u nj + j

(37)

Theorem 1 The explicit scheme (37) is consistent with the original equation (1) . Proof By Taylor series expansion, we have uxx =

u(x, t + k) − u(x, t) u 2 (x, t) + (u(x + h, t) + u(x − h, t)) φ1 2 γ − u(x, t) (u(x, t) + u(x + h, t)) + γ u(x, t) − u(x, t) u(x − h, t) 2 2 (u + ku t + k2 u tt + ··) − u = φ1 ≈ k   u2 h2 h2 u + hu x + u x x + u − hu x + u x x + ·· + 2 2 2 h2 h2 γ − u(u + u + hu x + u x x + ··) + γu − u(u − hu x + u x x + ··). 2 2 2 (38)

It follows that  k2 u2  u tt + · · + 2u + h 2 u x x + ·· 2 2 h2 uh 2 γ u x x + ·· (39) − u (2u + hu x + u x x + ··) + γu − u 2 + huu x + 2 2 2

u x x = ut −

Finally, we have

Comparative Study of Some Numerical Methods …

u x x = u t + γu − u 2 + u 3 − γu 2 −

105

k2 h2u2 h2u u tt + u x x + huu x + uxx 2 2 2 2 hγu h γu ux − uxx · · − 2 4 (40)

When k, h → 0, we recover the equation u x x = u t + γu − u 2 + u 3 − γu 2 = u t + u(1 − u)(γ − u).

(41) 

Numerical experiment for the explicit scheme We solve the numerical experiment using γ = 0.001, h = 0.1 for 0 ≤ x ≤ 1, and 0 ≤ x ≤ 10, for time, T = 1.0. Using the functional relationship, h = A2 k with γ = 0.001, h = 0.1, we obtain k = 1/14. We tabulate L 1 and L ∞ errors at some values of k in Tables 1 and 2 and observe that the method appears to be unstable at k = 1/14 ≈ 0.071. We next use γ = 0.5, h = 0.1 for 0 ≤ x ≤ 1 and 0 ≤ x ≤ 10. If we use the functional relationship, h = A2 k with γ = 0.5, h = 0.1 the value of k must be 0.09428, we use k = 1/11 ≈ 0.0909. The errors are displayed in Tables 3 and 4. − in Tables 1, 2, 3 and 4 indicate unbounded values. Tables 1, 2, 3 and 4, we deduce that functional relationship does not guarantee stability of the scheme. We study the stability analysis of the scheme using von Neumann stability analysis and method of freezing coefficient [32]. Using the ansatz, u nj = ξ n e I jw , we obtain the amplification factor of Eq. (37) as

Table 1 Computation of L 1 , L ∞ errors and CPU time using explicit scheme described by (37) from Namjoo and Zibaei [1] using γ = 0.001, h = 0.1, 0 ≤ x ≤ 1 at time T = 1.0 Time step (k) L 1 error L ∞ error CPU (s) 0.0005 0.001 0.002 0.004 0.005 0.008 0.016 0.032 1/14 ≈ 0.071 0.1

7.2345 × 10−13 7.1830 × 10−13 7.0800 × 10−13 6.8741 × 10−13 6.7711 × 10−13 − − − 1.2060 × 102 1.3581

1.0961× 10−12 1.0883× 10−12 1.0727× 10−12 1.0259 × 10−12 1.0259 × 10−12 − − − 2.1499 × 102 1.9441

1.122 0.811 0.700 0.624 0.662 − − − 0.585 0.550

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Table 2 Computation of L 1 , L ∞ errors and CPU time using explicit scheme described by (37) from Namjoo and Zibaei [1] using γ = 0.001, h = 0.1, 0 ≤ x ≤ 10 at time T = 1.0 Time step (k) L 1 error L ∞ error CPU (s) 7.4612 × 10−11 7.4084 × 10−11 7.3026 × 10−11 7.0911 × 10−11 6.9855 × 10−11 − − − 1.5906 ×102 1.3609

0.0005 0.001 0.002 0.004 0.005 0.008 0.016 0.032 1/14 ≈ 0.071 0.1

8.7850 × 10−12 8.7225 × 10−12 8.5975 × 10−12 8.3474 × 10−12 8.2223 × 10−12 − − − 1.6740 × 102 1.9447

3.180 1.802 1.273 0.971 0.846 − − − 0.686 0.644

Table 3 Computation of L 1 , L ∞ errors and CPU time using explicit scheme described by (37) from Namjoo and Zibaei [1] using γ = 0.5, h = 0.1, 0 ≤ x ≤ 1 at time T = 1.0 Time step (k) L 1 error L ∞ error CPU (s) 6.2098 × 10−5 6.1767 × 10−5 6.1104 × 10−5 5.9779 × 10−5 5.9117 × 10−5 − − − 1.5602 × 10131 1.8374 × 1049

0.0005 0.001 0.002 0.004 0.005 0.008 0.016 0.032 1/11 ≈ 0.0909 0.1

ξ=

9.4197 × 10−5 9.3694 × 10−5 9.2689 × 10−5 9.0680 × 10−5 8.9675 × 10−5 − − − 1.1393 × 10133 1.0756 × 1050

1.176 1.019 0.925 0.908 0.800 − − − 0.572 0.546

γ φ1 (2 cos(w) − 2) − φ1 u 2max cos(w) + φ1 u max (1 + cos(w)) ψ1 ψ2 2 γ

sin(w). + φ1 u max cos(w) + 1 − γφ1 − I φ1 u max 1 − 2

(42)

−2 A1 h

√

where ψ1 = 1−e2 A1 , ψ2 = e 2 A1−1 , A1 = 42 γ. We note that u max ≤ γ (from some numerical experiments which were carried out). 2 A1 h

First, we fix h = 0.1, γ = 0.001 and compute k as h/A2 . A plot of |ξ| against w ∈ [−π, π] is then obtained as depicted in Fig. 1a. We also consider h = 0.1 and γ = 0.5, and obtain plot of |ξ| against w ∈ [−π, π] in Fig. 1b. We observe that explicit scheme is not stable for all values of w ∈ [−π, π]. Hence, the explicit scheme constructed

Comparative Study of Some Numerical Methods …

107

Table 4 Computation of L 1 , L ∞ errors and CPU time using explicit scheme described by (37) from Namjoo and Zibaei [1] using γ = 0.5, h = 0.1, 0 ≤ x ≤ 10 at time T = 1.0 Time step (k) L 1 error L ∞ error CPU (s) 0.0005 0.001 0.002 0.004 0.005 0.008 0.016 0.032 1/11 ≈ 0.0909 0.1

6.5901 × 10−3 6.5544 × 10−3 6.4830 × 10−3 6.3403 × 10−3 6.2689 × 10−3 − − − 1.0623 × 10127 1.4743 × 1049

1.0134 × 10−3 1.0080 × 10−3 9.9703 × 10−4 9.7515 × 10−4 9.6422 × 10−4 − − − 9.4631 × 10127 8.3040 × 1049

3.723 2.176 1.661 1.025 0.854 − − − 0.564 0.552

Fig. 1 Plot of |ξ| against w using explicit scheme described by Eq. (37) from Namjoo and Zibaei [1]

described by Eq. (37) from Namjoo and Zibaei [1] using the exact solution is not useful scheme due to stability issues. An alternative way to understand why the explicit scheme is unstable for this numerical experiment We consider h = A2 k.

is explained√in the following lines: √ 4−2γ 2− 2γ 2 A1 and A1 = 4 γ, this gives A2 = 2 . Hence, hk = A12 = Since A2 = γ 2 √ , where γ 2− 2γ

∈ (0, 1). This gives 1 < hk < 2−2√2 and this can explain the instability of the explicit scheme. It can be shown that the condition for stability for a classical explicit scheme discretising Eq. (1) is hk2 < 21 . plots of exact and numerical profiles, vs x at time T = 1.0 are show in Fig. 2

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Fig. 2 Plot of u against x using explicit scheme described by Eq. (37) from Namjoo and Zibaei [1] at time T = 1.0 where x ∈ [0, 1] for a, c and x ∈ [0, 10] for b, d

5.2 Implicit Scheme If we choose u x x = ∂x ∂¯x u, this combination generates an implicit scheme. We have  u(x, t) − u(x − h, t) ∂x u(x, t) − ∂x u(x − h, t) , = ψ2 ψ2     2 A1 u(x, t) u(x − h, t) 2 A1 − . (43) u(x + h, t) 1 − u(x, t) 1 − = ψ2 γ ψ2 γ

∂x ∂¯x u = ∂x



It follows that 2 A1 2 A1 2 A1 u(x + h, t)u(x, t) 2 A1 u(x − h, t)u(x, t) − , ∂x ∂¯x u = u(x + h, t) − u(x, t) + ψ2 ψ2 γ ψ2 ψ2 γ

which gives

(44)

Comparative Study of Some Numerical Methods …

109

2 A1 2 A1 u(x, t) (u(x − h, t) − u(x + h, t)) . (u(x + h, t) − u(x, t)) + ψ2 γψ2 (45) We introduce A2 by adding and subtracting the expression ψA22 (u(x + h, t)− u(x, t)) into (45) and we have ∂x ∂¯x u =

2 A1 + A2 2 A1 ∂x ∂¯x u = u(x, t) (u(x − h, t) − u(x + h, t)) (u(x + h, t) − u(x, t)) + ψ2 γψ2 A2 − (u(x + h, t) − u(x, t)) . ψ2

(46) Since h = A2 k, it follows that

A2 ψ2

=

1 . φ2

Also u(x + h, t) = u(x, t − k). Therefore,

A2 u(x, t − k) − u(x, t) . (u(x + h, t) − u(x, t)) = ψ2 φ2

(47)

We thus have uxx =

u(x, t) − u(x, t − k) 2 A1 + A2 + (u(x + h, t) − u(x, t)) φ2 ψ2   u(x, t) − u(x + h, t) 2 A1 u(x − h, t) − u(x, t) , u(x, t) + + γ ψ2 ψ2

(48)

which can be expressed as uxx =

  u(x, t) − u(x, t − k) u(x + h, t) + 2 A1 (2 A1 + A2 )u(x, t) 1 − φ2 γ      u(x, t) u(x + h, t) 2 A1 − 2 A1 u(x, t) 1 − . + u(x, t) −2 A1 u(x − h, t) 1 − γ γ γ

(49) Further simplification gives uxx

and

  u(x, t) − u(x, t − k) u(x + h, t) = + γu(x, t) 1 − φ2 γ     γ u(x, t) γ 2 u(x + h, t) − u(x, t) 1 − u(x − h, t), − u (x, t) 1 − 2 γ 2 γ (50)

110

uxx =

K. M. Agbavon et al.

u(x, t) − u(x, t − k) + γu(x, t) − u(x, t)u(x + h, t) φ2 −

γ u 2 (x, t)u(x − h, t) γ 2 u 2 (x, t) u (x, t) + u(x + h, t) − u(x, t)u(x − h, t) + . 2 2 2 2

(51)

It follows uxx =

u(x, t) − u(x, t − k) u 2 (x, t) + (u(x + h, t) + u(x − h, t)) φ2 2 γ − u(x, t) (u(x, t) + u(x − h, t)) − u(x, t)u(x + h, t) + γu(x, t). (52) 2

The implicit scheme is therefore u nj+1 − 2u nj + u nj−1 ψ1 ψ2

=

u nj − u n−1 j φ2

 +

(u nj )2

u nj+1 + u nj−1 2



 −

(γu nj )

u nj + u nj−1



2 − u nj u nj+1 + γu nj (53)

which can be rewritten as  φ2 (u nj )2  n  φ2  n u j+1 + u nj−1 u j+1 − 2u nj + u nj−1 + ψ1 ψ2 2 u nj  n  u j + u nj−1 − φ2 u nj u nj+1 + φ2 γ u nj . − φ2 γ 2

= u nj − u n−1 j

(54)

Theorem 2 The implicit scheme (54) is consistent with the original equation (1) . Proof The proof is similar as in the case of an explicit scheme by using Taylor expansions. 

Numerical experiment for the implicit scheme The amplification factor of Eq. (54) can be obtained from the following: ξ −1 = 1 −

γ φ2 − 1 sin(w) (2 cos(w) − 2) + φ2 u 2max cos(w) + I φ2 u max ψ1 ψ2 2

γ γ + 1 cos(w) − φ2 u max + φ2 γ. − φ2 u max 2 2 (55)

Comparative Study of Some Numerical Methods …

111

Plots of ξ against w ∈ [−π, π] for the two cases: h = 0.1, γ = 0.001 and h = 0.1, γ = 0.5 are shown in Figs. 3a and 3b and we can deduce that the scheme is stable in both cases as |ξ| ≤ 1 for w ∈ [−π, π] . Here, since we don’t have any stability issues with implicit scheme, we can choose k, such that h = A2 k. This can be deduced from Fig. 3. L 1 and L 00 errors are using implicit scheme for the four cases are displayed in (Tables 5, 6, 7 and 8). Plots of profiles vs x at time T = 1.0 displayed in Fig. 4

Fig. 3 Plot of |ξ| against w using implicit scheme described by (54) from Namjoo and Zibaei [1]

Table 5 Computation of L 1 and L ∞ errors using implicit scheme described by (54) from Namjoo and Zibaei [1] with γ = 0.001, h = 0.1, 0 ≤ x ≤ 1 at time T = 1.0 Time step (k) L 1 error L ∞ error 0.0005 0.001 0.002 0.004 0.005 0.008 1/14 ≈ 0.071 0.1

7.2041 × 10−11 6.2247 × 10−11 7.0493 × 10−11 6.8431 × 10−10 6.7399 × 10−10 6.4308 × 10−10 4.8835 × 10−12 2.9959 × 10−11

8.7556 × 10−12 7.5695 × 10−12 8.5683 × 10−12 8.3186 × 10−12 8.1937 × 10−12 7.8192 × 10−12 2.2035 × 10−12 3.6599 × 10−12

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Table 6 Computation of L 1 and L ∞ errors using implicit scheme described by (54) from Namjoo and Zibaei [1] with γ = 0.001, h = 0.1, 0 ≤ x ≤ 10 at time T = 1.0 Time step (k) L 1 error L ∞ error 0.0005 0.001 0.002 0.004 0.005 0.008 1/14 ≈ 0.071 0.1

8.1354 × 10−10 7.6458 × 10−10 8.6291 × 10−10 8.3833 × 10−10 8.2604 × 10−10 7.8916 × 10−10 1.2821 × 10−11 3.4110 × 10−10

9.1804 × 10−12 7.9903 × 10−12 8.9924 × 10−12 8.7419 × 10−12 8.6166 × 10−11 8.2408 × 10−12 2.9990 × 10−13 3.6628 × 10−12

Table 7 Computation of L 1 and L ∞ errors using implicit scheme described by (54) from Namjoo and Zibaei [1] with γ = 0.5, h = 0.1, 0 ≤ x ≤ 1 at time T = 1.0 Time step (k) L 1 error L ∞ error 0.0005 0.001 0.002 0.004 0.005 0.008 1/11 ≈ 0.0909 0.1

6.6849 × 10−3 8.2833 × 10−3 6.6065 × 10−3 6.5018 × 10−3 6.4495 × 10−3 6.2923 × 10−3 4.4861 × 10−4 4.5716 × 10−3

1.3886 × 10−3 1.3837 × 10−3 1.3739 × 10−3 1.3545 × 10−3 1.3447 × 10−3 1.3154 × 10−3 9.1301 × 10−4 9.9875 × 10−4

Table 8 Computation of L 1 and L ∞ errors using implicit scheme described by (54) from Namjoo and Zibaei [1] with γ = 0.5, h = 0.1, 0 ≤ x ≤ 10 at time T = 1.0 Time step (k) L 1 error L ∞ error 0.0005 0.001 0.002 0.004 0.005 0.008 1/11 ≈ 0.0909 0.1

4.8991 × 10−1 4.8961 × 10−1 4.8901 × 10−1 4.8780 × 10−1 4.8719 × 10−1 4.8537 × 10−1 4.3641×10−1 4.3118 × 10−1

7.5539 × 10−3 7.5500 × 10−3 7.5423 × 10−3 7.5266 × 10−3 7.5188 × 10−3 7.4953 × 10−3 6.8410 × 10−3 6.7691 × 10−3

Comparative Study of Some Numerical Methods …

113

Fig. 4 Plot of u against u using implicit scheme described Eq. (54) from Namjoo and Zibaei [1] at time, T = 1.0 where x ∈ [0, 1] for a, c and x ∈ [0, 10] for b, d

6 Nonstandard Finite Difference Scheme (NSFD) We present a derivation of two versions of NSFD Schemes which can be named as NSFD1 and NSFD2 [1]. NSFD1 is improved to generate NSFD3 scheme.

6.1 NSFD1 Scheme We note that the right hand side of Eq. (1) is −u 3 + (1 + γ)u 2 − γu. We use the following discrete approximations for the right hand of Eq. (1): 2  3 3 1  n 3 u ≈ − u nj−1 u n+1 + . (56) − γu(x j , tn ) ≈ −γu n+1 j , − u(x j , tn ) j 2 2 j−1

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This gives the following scheme: − u nj u n+1 j φ1

where ψ1 = scheme is

−

u nj+1 − 2u nj + u nj−1 ψ1 ψ2

1−e−2 A1 h , 2 A1

ψ2 =

e2 A1 h −1 2 A1

=−

3  n 2 n+1 1  n 3 u u uj + 2 j−1 2 j−1  2 + (1 + γ) u nj−1 − γu n+1 j ,

and φ1 =

1−e−2 A1 A2 k . 2 A1 A2

A single expression for the

 (1 − = u n+1 j

2R)u nj

+

R(u nj+1

+

u nj−1 )

(57)

+ φ1 (1 + γ)



u nj−1

2



3  n u j−1 + 1 2



2 1 + φ1 γ + 23 φ1 u nj−1

,

(58) where R =

φ1 . ψ1 ψ2

Theorem 3 If 1 − 2R ≥ 0, the numerical solution of Eq. (1) satisfies 0 ≤ u nj ≤ 1 =⇒ 0 ≤ u n+1 ≤ 1, j and the dynamical consistency holds for all relevant values of n and j. Remark 1 As stated in the introduction, the concept of nonstandard finite difference required dynamical consistency (positivity, boundedness, preservation of fixed points) which helps to avoid numerical instabilities. The fixed points of Eq. (1) are u ∗1 = 0, u ∗2 = 1 (which are stable) and u ∗3 = γ which is unstable. Furthermore, Roger and Mickens [33] showed preservation of local stabilities of all fixed points. Proof For positivity of the scheme given by Eq. (58), we have u n+1 ≥ 0 if only j

2 3 n n 1 − 2R ≥ 0, since u j ≥ 0 by assumption and 1 + φ1 γ + 2 φ1 u j−1 > 0. To obtain the condition for positivity of NSFD1, we solve R = ψφ1 ψ1 2 ≤ 21 which implies that, by replacing φ1 , ψ1 , ψ2 , with their respective expressions, we obtain 

1 − e−2 A1 A2 k 2 A1 A2



2 A1 1 − e−2 A1 h



2 A1 2 A e 1h − 1

 ≤

1 , 2

(59)

which gives   1 A2 (e2 A1 h − 1)2 , k≤− ln 1 − 2 A1 A2 4 A1 e 2 A1 h

(60)

Comparative Study of Some Numerical Methods …

115

and finally ⎡ k≤−

2 ⎢ ln ⎣1 − γ(2 − γ)



√

2 ⎤  e 22 γh − 1 2−γ ⎥ √ ⎦. 2 γh 2γ 2 e

(61)

We assume that 0 ≤ u nj ≤ γ. If the scheme is bounded, we need to prove that 0 ≤ ≤ γ or u n+1 − γ ≤ 0. Consider u n+1 j j 

(u n+1 j

2 3  − γ) 1 + φ1 γ + φ1 u nj−1 2

 = (1 − 2R)u nj + R(u nj+1 + u nj−1 )    n 2 1  n 3 u + φ1 (1 + γ) u j−1 + 2 j−1 2 3  (62) − γ − φ1 γ 2 − φ1 u nj−1 . 2



3

2

2 It follows u nj−1 = u nj−1 u nj−1 ≤ γ u nj−1 since 0 ≤ u nj ≤ γ for all values for n and j. Therefore, 

(u n+1 j

2 3  − γ) 1 + φ1 γ + φ1 u nj−1 2

 ≤ (1 − 2R)γ + 2γ R  2 γ  n 2

+ φ1 γ 2 + φ1 γ u nj−1 + u 2 j−1 2 3  − γ − φ1 γ 2 − φ1 u nj−1 = 0. (63) 2

We therefore obtain u n+1 − γ ≤ 0. Hence, NSFD1 satisfies boundedness j properties.  For γ = 0.001 and h = 0.1 in Eq. (61) gives k ≤ 5.0000 × 10−3 , while for γ = 0.5 and h = 0.1, we get k ≤ 5.0052 × 10−3 . We tabulate L 1 L ∞ errors as well as CPU time and rate of convergence with respect to time (using L 1 error) for the four cases in Tables 9–12. We also obtain plot of u against x at time, T = 1.0 in Fig. 5. From Tables 9, 10, 11 and 12, we deduce that though the L 1 , L ∞ are of the order 10−12 , 10−10 for cases 1, 2 and of order 10−4 , 10−2 for cases 3, 4, the rate of convergence with respect to time (and L 1 error) is negative and this indicates that NSFD1 has convergence issues. This could be due to approximations of −γu(x j , tn )

2

3  3 n+1 3 1 n n u u and approximation of − u(x , t ) by − u + by −γu n+1 j n j−1 j−1 , j j 2 2 where u nj−1 and u n+1 are both non-local approximations. (1 + γ) u 2 can be approxj

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Fig. 5 Plot of u against x using NSFD1 scheme at time T = 1.0, where x ∈ [0, 1], for a, c and x ∈ [0, 10] for b, d Table 9 Computation of L 1 and L ∞ errors, CPU time and rate of convergence in time using NSFD1 with γ = 0.001, h = 0.1, 0 ≤ x ≤ 1 at time, T = 1.0 Time step (k) L 1 error L ∞ error Rate of CPU (s) convergence 0.005 0.004 0.002 0.001 0.0005

1.3053 × 10−12 1.3362 × 10−12 1.3981 × 10−12 1.4290 × 10−12 1.4444 × 10−12

1.9777 × 10−12 2.0246 × 10−12 2.1183 × 10−12 2.1651 × 10−12 2.1885 × 10−12

− −0.1049 −0.0653 −0.0315 −0.0154

0.561 0.588 0.588 0.688 0.745

2  3 imated by (1 + γ) u nj+1 u nj−1 or (1 + γ) u nj . − u(x j , tn ) can be approximated 2 3 1 u nj . by − 23 u nj u n+1 + j 2

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Table 10 Computation of L 1 and L ∞ errors, CPU time and rate of convergence in time using NSFD1 with γ = 0.001, h = 0.1, 0 ≤ x ≤ 10 at time, T = 1.0 Time step (k) L 1 error L ∞ error Rate of CPU (s) convergence 0.005 0.004 0.002 0.001 0.0005

1.3468 × 10−10 1.3786 × 10−10 1.4421 × 10−10 1.4739 × 10−10 1.4897 × 10−10

1.5853 × 10−11 1.6228 × 10−11 1.6978 × 10−11 1.7353 × 10−11 1.7540 × 10−11

− −0.1046 −0.0649 −0.0315 −0.0154

0.772 0.828 1.118 1.561 2.991

Table 11 Computation of L 1 and L ∞ errors CPU time and rate of convergence in time using NSFD1 with γ = 0.5, h = 0.1, 0 ≤ x ≤ 1 at time, T = 1.0 Time step (k) L 1 error L ∞ error Rate of CPU (s) convergence 0.005 0.004 0.002 0.001 0.0005

2.6024 × 10−4 2.6311 × 10−4 2.6886 × 10−4 2.7173 × 10−4 2.7317 × 10−4

3.9470 × 10−4 3.9905 × 10−4 4.0776 × 10−4 4.1212 × 10−4 4.1429 × 10−4

− -0.0491 -0.0312 -0.0153 -7.6250 × 10−3

0.565 0.569 0.645 0.685 0.829

Table 12 Computation of L 1 and L ∞ errors, CPU time and rate of convergence using NSFD1 with γ = 0.5, h = 0.1, 0 ≤ x ≤ 10 at time, T = 1.0 Time step (k) L 1 error L ∞ error Rate of CPU (s) convergence 0.005 0.004 0.002 0.001 0.0005

2.9989 × 10−2 3.0267 × 10−2 3.0825 × 10−2 3.1104 × 10−2 3.1244 × 10−2

4.5242 × 10−3 4.5665 × 10−3 4.6515 × 10−3 4.6941 × 10−3 4.7154 × 10−3

− −0.0413 −0.0264 −0.0129 −6.4790× 10−3

0.743 0.749 1.057 1.610 2.775

6.2 NSFD2 Scheme Here, we use the following approximations for the right-hand side of Eq. (1) :  2 n u(x j , tn ) ≈ u n+1 u(x j , tn ) ≈ u n+1 j , j u j,    3 2 u nj . u(x j , tn ) ≈ u n+1 j The NSFD2 scheme when used to discretise Eq. (1) is given by

(64) (65)

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u n+1 − u nj j φ1

−

u nj+1 − 2u nj + u nj−1 ψ1 ψ2

 n 2 n+1 n u j + (1 + γ)u n+1 = −u n+1 j j u j − γu j . (66)

or = u n+1 j

(1 − 2R)u nj + R(u nj+1 + u nj−1 ) φ1 . 2 , where R = ψ1 ψ2 1 + φ1 γ − (1 + γ)φ1 u nj + φ1 u nj

(67)

Theorem 4 If 1 − 2R ≥ 0 and 1 − φ1 γ 2 ≥ 0, the numerical solution of Eq. (1) satisfies ≤ 1, 0 ≤ u nj ≤ 1 =⇒ 0 ≤ u n+1 j and the dynamical consistency holds for all relevant values of n and j. Proof For positivity of NSFD2, we have u n+1 ≥ 0 if only 1 − 2R ≥ 0 and 1 − j n 2 φ1 γ ≥ 0. The coefficient of u j must be non-negative. We also need to have  2 1 + φ1 γ − (1 + γ)φ1 u nj + φ1 u nj > 0. We note that 0 ≤ u nj ≤ γ. Hence,    2  2 1 − (1 + γ)φ1 u nj − φ1 γ + φ1 u nj ≥ 1 + γφ1 − γφ1 − γ 2 φ1 + φ1 u nj  2 ≥ 1 − φ1 γ 2 + φ1 u nj ≥ 1 − φ1 γ 2 . To obtain the condition for positivity of NSFD2, we solve R = 1 − φ1 γ 2 > 0. Solving R ≤ 21 gives ⎡ k≤−

2 ⎢ ln ⎣1 − γ(2 − γ)



2 ⎤ √  e 22 γh − 1 2−γ ⎥ √ ⎦. 2 γh 2γ e2

(68) φ1 ψ1 ψ2

≤

1 2

and

(69)

We also need to solve  1−

1 − e−2 A1 A2 k A1 A2

 γ 2 > 0,

(70)

and this gives k≤−

 γ 2 ln 1 − (2 − γ) . γ(2 − γ) 4

(71)

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and finally, we obtain ⎧   2 − γ(2−γ) ln 1 − γ4 (2 − γ) , ⎪ ⎪ ⎡  √ 2 ⎤ ⎨

e 22 γh −1 k≤ 2−γ 2 ⎣ ⎦. ⎪ √ ⎪ 2 ⎩− γ(2−γ) ln 1 − 2γ e

(72)

2 γh

For boundedness of NSFD2, we assume that 0 ≤ u nj ≤ γ. Consider   n + φ u n 2 = (1 − 2R)u n + R(u n n (u n+1 − γ) 1 + φ γ − (1 + γ)φ u 1 1 j 1 j j j+1 + u j−1 ) j − γ − γ 2 φ1 + γ(1 + γ)φ1 u nj 2 − γφ1 u nj . (73)

It follows, since 0 ≤ u nj ≤ γ for all values for n and j,  2

(u n+1 ≤ (1 − 2R)γ + 2γ R − γ) 1 + φ1 γ − (1 + γ)φ1 u nj + φ1 u nj j − γ − γ 2 φ1 + γ(1 + γ)φ1 u nj  2 − γφ1 u nj . (74) Finally, we have  

  n + φ un 2 ≤ − φ γ n 2 − γu n − u n + γ u (u n+1 − γ) 1 + φ γ − (1 + γ)φ u 1 1 1 1 j j j j j j = −φ1 γ(u nj − 1)(u nj − γ). (75)

Since 0 ≤ u nj ≤ γ and γ ∈ (0, 1), we have (u nj − 1)(u nj − γ) ≥ 0 and  2

(u n+1 ≤ 0. − γ) 1 + φ1 γ − (1 + γ)φ1 u nj + φ1 u nj j Therefore we have 0 ≤ u n+1 ≤ γ. Hence, the boundedness of NSFD2. j

(76) 

Solving (72) for γ = 0.001, h = 0.1, gives k ≤ 5.0000 × 10−3 and k ≤ 5.0012 × 10 When we solve for γ = 0.5, h = 0.1, we obtain k ≤ 5.0052 × 10−3 , and k ≤ 5.5370 × 10−1 . Hence, NSFD2 scheme is positive definite if −1

(i) k ≤ 5.0000 × 10−3 when γ = 0.001. (ii) k ≤ 5.0052 × 10−3 when γ = 0.5. We tabulate L 1 , L ∞ errors and CPU time and rate of convergence in time in Tables 13, 14, 15 and 16 and obtain plots of u against x in Fig. 6.

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Table 13 Computation of L 1 and L ∞ errors, CPU time and rate of convergence using NSFD2 with γ = 0.001, h = 0.1, 0 ≤ x ≤ 1 at time T = 1.0 Time step (k) L 1 error L ∞ error Rate of CPU (s) convergence 0.005 0.004 0.002 0.001 0.0005

1.0302 × 10−13 8.2413 × 10−14 4.1207 × 10−14 2.0598 × 10−14 1.0303 × 10−14

1.5608 × 10−13 1.2487 × 10−13 6.2435 × 10−14 3.1209 × 10−14 1.5611 × 10−14

− 1.000 1.000 1.000 0.999

0.498 0.514 0.562 0.590 0.685

Table 14 Computation of L 1 and L ∞ errors, CPU time and rate of convergence using NSFD2 with γ = 0.001, h = 0.1, 0 ≤ x ≤ 10 at time T = 1.0 Time step (k) L 1 error L ∞ error Rate of CPU (s) convergence 0.005 0.004 0.002 0.001 0.0005

1.0609 × 10−11 8.4863 × 10−12 4.2427 × 10−12 2.1206 × 10−12 1.0607 × 10−12

1.2486 × 10−12 9.9890 × 10−13 4.9945 × 10−13 2.4966 × 10−13 1.2488 × 10−13

− 1.000 1.000 1.000 0.999

0.682 0.746 1.014 1.485 2.531

Table 15 Computation of L 1 and L ∞ errors, CPU time and rate of convergence using NSFD2 with γ = 0.5, h = 0.1, 0 ≤ x ≤ 1 at time T = 1.0 Time step (k) L 1 error L ∞ error Rate of CPU (s) convergence 0.005 0.004 0.002 0.001 0.0005

7.2879 × 10−6 5.8241 × 10−6 2.8964 × 10−6 1.4325 × 10−6 7.0051 × 10−7

1.1046 × 10−5 8.8276 × 10−6 4.3902 × 10−6 2.1715 × 10−6 1.0620 × 10−6

− 1.005 1.008 1.016 1.032

0.611 0.668 0.678 0.724 0.764

Table 16 Computation of L 1 and L ∞ errors, CPU time and rate of convergence using NSFD2 with γ = 0.5, h = 0.1, 0 ≤ x ≤ 10 at time T = 1.0 Time step (k) L 1 error L ∞ error Rate of CPU (s) convergence 0.005 0.004 0.002 0.001 0.0005

4.6563 × 10−4 3.7403 × 10−4 1.9066 × 10−4 9.8889 × 10−5 5.2984 × 10−5

7.9312 × 10−5 6.3679 × 10−5 3.2398 × 10−5 1.6760 × 10−5 8.9589 × 10−6

− 0.9817 0.9722 0.9471 0.9003

0.696 0.751 1.070 1.372 2.515

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Fig. 6 Plot of u against x using NSFD2 scheme at time T = 1.0, where x ∈ [0, 1] for a, c and x ∈ [0, 10] for b, d

6.3 NSFD3 Scheme The NSFD1 scheme has convergence issues. We present a slight modification of NSFD1 and we baptise the new scheme as NSFD3. We propose the following: − u nj u n+1 j φ1

where ψ1 = scheme is

−

1−e−2 A1 h , 2 A1

u nj+1 − 2u nj + u nj−1 ψ1 ψ2

ψ2 =

e2 A1 h −1 2 A1

=−

and φ1 =

3  n 2 n+1 1  n 3 u uj + u 2 j 2 j  2 + (1 + γ) u nj − γu n+1 j ,

1−e−2 A1 A2 k . 2 A1 A2

(77)

A single expression for the

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u n+1 = j

 2 3  (1 − 2R)u nj + R(u nj+1 + u nj−1 ) + φ1 (1 + γ) u nj + 21 u nj 2 1 + φ1 γ + 23 φ1 u nj

where R =

, (78)

φ1 . ψ1 ψ2

Theorem 5 If 1 − 2R ≥ 0, the numerical solution of Eq. (1) satisfies 0 ≤ u nj ≤ 1 =⇒ 0 ≤ u n+1 ≤ 1, j and the dynamical consistency holds for all relevant values of n and j. 2 Proof For positivity, we should have 1 − 2R ≥ 0 and 1 + φ1 γ + 23 φ1 u nj > 0,since u nj ≥ 0. To obtain the condition for positivity of NSFD3, we solve R = ψφ1 ψ1 2 ≤ 21 which is similar condition for positivity of NSFD1. Therefore, the condition is ⎡ k≤−

2 ⎢ ln ⎣1 − γ(2 − γ)



2 ⎤ √  e 22 γh − 1 2−γ ⎥ √ ⎦. 2 γh 2γ e2

(79)

We assume that 0 ≤ u nj ≤ γ. For the boundedness, we need to prove that 0 ≤ ≤ γ. We consider u n+1 j 

(u n+1 j

3  2 − γ) 1 + φ1 γ + φ1 u nj 2

 = (1 − 2R)u nj + R(u nj+1 + u nj−1 )    n 2 1  n 3 u + φ1 (1 + γ) u j + 2 j 3  2 (80) − γ − φ1 γ 2 − φ1 u nj . 2

Since 0 ≤ u nj ≤ γ for all values for n and j, we have 

(u n+1 j

3  2 − γ) 1 + φ1 γ + φ1 u nj 2

 ≤ (1 − −2R)γ + 2γ R   γ  n 2

2 + φ1 γ 2 + φ1 γ u nj + u 2 j 3  2 (81) − γ − φ1 γ 2 − φ1 u nj = 0. 2

Hence, u n+1 ≤ γ. It follows that NSFD3 satisfies boundedness properties. j



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Table 17 Computation of L 1 and L ∞ errors, CPU time and rate of convergence in time using NSFD3 with γ = 0.001, h = 0.1, 0 ≤ x ≤ 1 at time T = 1.0 Time step (k) L 1 error L ∞ error Rate of CPU (s) convergence 0.005 0.004 0.002 0.001 0.0005

1.5460 × 10−13 1.2368 × 10−13 6.1841 × 10−14 3.0915 × 10−14 1.5462 × 10−14

2.3424 × 10−13 1.8739 × 10−13 9.3698 × 10−14 4.6841 × 10−14 2.3426 × 10−14

− 1.000 0.999 1.000 0.999

0.770 0.920 0.928 1.092 1.215

Table 18 Computation of L 1 and L ∞ errors, CPU time and rate of convergence in time using NSFD3 with γ = 0.001, h = 0.1, 0 ≤ x ≤ 10 at time T = 1.0 Time step (k) L 1 error L ∞ error Rate of CPU (s) convergence 0.005 0.004 0.002 0.001 0.0005

1.5930 × 10−11 1.2743 × 10−11 6.3709 × 10−12 3.1847 × 10−12 1.5927 × 10−12

1.8750 × 10−12 1.4910 × 10−12 7.500 × 10−13 3.7494 × 10−13 1.8752 × 10−13

− 1.000 1.000 1.000 0.999

0.947 1.100 1.296 1.983 2.922

Table 19 Computation of L 1 and L ∞ errors, CPU time and rate of convergence in time using NSFD3 with γ = 0.5, h = 0.1, 0 ≤ x ≤ 1 at time T = 1.0 Time step (k) L 1 error L ∞ error Rate of CPU (s) convergence 0.005 0.004 0.002 0.001 0.0005

1.4440 × 10−5 1.1546 × 10−5 5.7572 × 10−6 2.8628 × 10−6 1.4157 × 10−6

2.1896 × 10−5 1.7508 × 10−5 8.7301 × 10−6 4.3413 × 10−6 2.1469 × 10−6

− 1.002 1.004 1.007 1.016

0.707 0.892 0.934 1.043 1.093

For γ = 0.001 and h = 0.1 in Eq. (79) gives k ≤ 5.0000 × 10−3 , while for γ = 0.5 and h = 0.1, we get k ≤ 5.0052 × 10−3 . We tabulate L 1 , L ∞ errors as well as CPU time and rate of convergence with respect to time (using L 1 error) for the four cases in Tables 17, 18, 19 and 20. We also obtain plot of u against x at time T = 1.0 in Fig. 7. The scheme proposed is quite effective for all the four cases.

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Table 20 Computation of L 1 and L ∞ errors, CPU time and rate of convergence using NSFD3 with γ = 0.5, h = 0.1, 0 ≤ x ≤ 10 at time T = 1.0 Time step (k) L 1 error L ∞ error Rate of CPU (s) convergence 0.005 0.004 0.002 0.001 0.0005

1.2452 × 10−3 9.9846 × 10−4 5.0366 × 10−4 2.5559 × 10−4 1.3138 × 10−4

1.9632 × 10−4 1.5743 × 10−4 7.9464 × 10−5 4.0389 × 10−5 2.0838 × 10−5

− 0.989 0.987 0.978 0.960

0.890 0.749 1.335 1.933 2.775

Fig. 7 Plot of u against x using NSFD3 scheme at time T = 1.0, where x ∈ [0, 1] for a, c and x ∈ [0, 10] for b, d

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7 Conclusion In this work, we use four numerical schemes from [1] to solve FitzHugh–Nagumo equation with specified initial and boundary conditions. Two of these schemes (explicit and implicit) are constructed from the exact solution and the other two are non-standard finite schemes which are named as NSFD1, NSFD2. Some highlights of this work as compared to the work in [1] are described below. Firstly, we consider another value of γ besides the value 0.001 and we also work with short and long domains. We observe that the results are much better when γ = 0.001 as compared to γ = 0.5. Secondly, we test the two schemes constructed by Namjoo and Zibaei from the exact solution and demonstrate that the explicit scheme has stability issues using numerical experiments and thorough analysis of the stability. The implicit scheme is quite effective and works when h = A2 k. Thirdly, NSFD1 has convergence issues due to its rate of convergence result. A modification of NSFD1 is performed to generate NSFD3. NSFD2 and NSFD3 schemes are quite effective methods to solve the FitzHugh–Nagumo equation with reasonable L 1 , L ∞ errors and the rate of convergence in time from numerical experiment are in agreement with theoretical rate of convergence in time. Acknowledgements The authors are very grateful to the two anonymous reviews for their comments and suggestions which helped to improve the paper considerably.

Disclosure Statement None of the authors has competing interests.

Funding A. R. Appadu is grateful to the National Research Foundation of South Africa, grant number 95864 for funding. Mr. K. M. Agbavon has been supported by a Ph.D. bursary from DST/NRF SARCHI Chair on Mathematical Models and Methods in Bioengineering and Biosciences (M 3 B 2 ) grant 82770 to carry out this research (from 2016 to 2018). Most of the work was done by K. M. Agbavon under supervision of A. R. Appadu. B. Inan carried out coding and presented the result for the implicit scheme.

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Analytical Solution of Neutron Diffusion Equation in Reflected Reactors Using Modified Differential Transform Method Mohammed Shqair and Essam R. El-Zahar

Abstract In this paper, the analytical solution of neutron diffusion equation in reflected reactors is obtained using Modified Differential Transform Method (MDTM). The MDTM is applied successfully on singular and non-singular initial value problems arising for the essential reactor geometries. Here, the reactors will not only consist of fuel part (bare reactors) but also it has a core and reflected parts (reflected reactors). A comparison with results in literature and transport theory data is presented. The results confirm that the MDTM is effective and reliable in solving the considered problems.

1 Introduction The Modified Differential Transform Method (MDTM) a semi-analytical method which is as an alternative of existing methods for solving different geometries reflected reactors neutron diffusion equation. Differential Transform Method (DTM) is introduced by Zhou in a study of electric circuits [1]. This method is a formalised modified version of Taylor series method where the derivatives are evaluated through recurrence relations and not symbolically as the traditional Taylor series method. This method has been used effectively to obtain highly accurate solutions for large classes of linear and nonlinear problems (see, e.g. [1–9]). There is no need for discretisation, perturbations and further large computational work and round-off errors are avoided. M. Shqair (B) · E. R. El-Zahar Department of Physics, College of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia e-mail: [email protected] E. R. El-Zahar e-mail: [email protected] E. R. El-Zahar Faculty of Engineering, Department of Basic Engineering Science, Menofia University, Shebin El-Kom 32511, Egypt © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 D. Zeidan et al. (eds.), Computational Mathematics and Applications, Forum for Interdisciplinary Mathematics, https://doi.org/10.1007/978-981-15-8498-5_6

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Additionally, DTM does not generate secular terms (noise terms) and does not need analytical integrations as other semi-analytical methods like homotopy perturbation method (HPM) [10], homotopy analysis method (HAM) [11], adomian decomposition method (ADM) [12] or variational iteration method (VIM) [13], where the methodologies of these methods are described in their corresponding references, the DTM is an attractive tool for solving differential equations. Since the considered reactor problems here are described by second-order homogenous differential equations with ordinary or regular (removable) singular points with bounded and analytic solutions then the MDTM [9] can be applied effectively to get the power series solutions of these problems. Different classes of neutron diffusion problems are solved using different solutions, Cassell and Williams used homotopy perturbation method to solve the neutron diffusion equation for a hemisphere with mixed boundary conditions [14]. Khasawneh, Dababneh and Odibat solved the neutron diffusion equation in hemispherical symmetry using the homotopy perturbation method [15]. Dababneh, Khasawneh and Odibat applied an alternative solution of the neutron diffusion equation in cylindrical symmetry [16]. Shqair, El-Ajou and Nairat used Residual Power Series Method to solve Multi-Energy Groups of Neutron Diffusion Equations [17]. Shqair solved different geometries reflected reactors neutron diffusion equation using the homotopy perturbation method [10]. Moreover, Shqair developed a new approach of homotopy perturbation method to solve the two-group reflected cylindrical reactor [18]. Momani and Odibat find the analytical solution of a time-fractional Navier–Stokes equation by Adomian decomposition method [12]. Dianchen Lu and Jie Liu applied the homotopy analysis method for solving the variable coefficient KdV–Burgers equation [11]. Abdul-Majid Wazwaz used the variational iteration method for solving linear and nonlinear ODEs and scientific models with variable coefficients [13]. Ghasemi, Hatami and Ganji used Differential Transformation Method for solving the nonlinear temperature distribution equation in a longitudinal fin with temperature-dependent internal heat generation and thermal conductivity [19], Nairat, Shqair and Alhalholy solved cylindrically symmetric fractional Helmholtz equation and applied the solution in optics and diffusion equation [20] and other methods [21–23]. In this paper, the description of the MDTM will be discussed in Sect. 2. The application of MDTM in nuclear reactor theory equations will be presented for both spherical and slab reactors in Sect. 3. We will apply the MDTM results on numerical examples and compare these results with other methods and transport theory data in Sect. 4.

2 Description of the Method Many real-life phenomena in various fields of science and engineering are solved using approximation methods and the others are still need to be solved or to improve their approximate solutions. There are several approximation methods that have been developed to deal with these problems [1–9]. Here, the MDTM that has been

Analytical Solution of Neutron Diffusion Equation …

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developed for the analytical solution of ODEs is presented in this section for the considered second order homogenous initial value problem (IVP) a(x)ϕ (x) + p(x)ϕ (x) + q(x)ϕ(x) = 0,

ϕ(x0 ) = α, ϕ (x0 ) = β,

(1)

where the a(x), p(x) and q(x) are analytic functions, α and β are constants and ϕ(x) is a bounded solution of Eq. (1). Let [x0 , T ] be the interval over which we want to find the solution of the IVP Eq. (1). In actual applications of the MDTM, the Nth-order approximate solution of the IVP Eq. (1) can be expressed by the finite series N  (k) (x − x0 )k , x ∈ [x0 , T ], (2) ϕ(x) = k=0

where (k) =

  1 d k ϕ(x) . k! d x k x=x0

(3)

Using some fundamental properties of MDTM, (Table 1), the IVP Eq. (1) can be transformed into the following recurrence relation: k 

(A (i) (k + 2 − i) (k + 1 − i)  (k + 2 − i) + P (i) ((k + 1 − i)

i=0

(4)

 (k + 1 − i)) + Q (i)  (k − i)) = 0,  (0) = α,  (1) = β,

where A, P, Q and  are the differential transforms of the functions a(x), p(x), q(x) and ϕ(x), respectively. In this study, a(x), p(x) and q(x) are polynomials and so the

Table 1 Fundamental operations of differential transform method Original function Transformed function y (x) = β (u (x) ± v (x)) y (x) = u (x) v (x) m y (x) = d d xu(x) m y (x) = x m y (x) = x m u (x)

Y (k) = β U (k) ± β V (k)  Y (k) = k=0 U () V (k − ) Y (k) = (k+m)! k! U (k +m) 1; Y (k) = δ(k − m) = 0; Y (k) = U (k − m), k ≥ m Y (k) =

λk k!

y (x) = sin (ωx)

Y (k) =

ωk k!

y (x) = cos (ωx)

Y (k) =

ωk k!

y (x) =

eλx

if k = m i f k = m

sin( kπ 2 )

=

⎧ ⎨0 ;

k ∈ even k−1

⎩ ωk (−1) 2 ; k! ⎧ ⎨ ωk (−1) k2 ; k! cos( kπ 2 )=⎩ 0;

k ∈ odd k ∈ even k ∈ odd

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following MDTM property Eq. (5) can be applied directly to Eq. (1) and by solving the result recurrence relation, the differential transform (k), k ≥ m can be easily obtained (5) If u (x) = x m y(x) then U (k) = Y (k − m), k ≥ m, where U (k) and Y (k) are the differential transforms of the functions u(x) and y(x), respectively.

3 Theory The nuclear reactor theory (which is a branch of nuclear physics that specialises in studying the nuclear reactors mechanism and their parts) is very important in supplying the energy to the civilisation to perform its tasks. For 100 years, the nuclear reactor theory takes high interest from the scientist all over the world. The behaviour of neutrons in the nuclear reactor parts is important issues in the nuclear reactor theory, this behaviour can be characterised by the neutron diffusion equation. For this purpose, there are many published papers that interested in solving one part (bare reactors) of neutron diffusion equations [14–17]. Recently, Shqair developed new techniques for solving reactors consist of two parts (core and reflected parts) which can be considered as the most essential reactor parts [10, 18]. In this paper, we consider a two-part reactor and solve the neutron diffusion equation for two different geometries. The neutron diffusion equation which is considered as a special case of neutron transport equation after applying Fick’s law (this law contains conditions which make transport equation more applicable mathematically). In order to study the motion of neutrons, or neutron diffusion, it is assumed that neutrons tend to diffuse from regions of high neutron flux to low one. This can be represented by the continuity equation: 1 ∂φ (r, t) = s (r, t) −a φ (r, t) −∇.J (r, t) . v ∂t

(6)

Equation (6) gives the relation between both neutron flux φ(r, t) and neutron current density J (r, t), where v and a constants are the neutrons velocity and macroscopic cross section, respectively, and s (r, t) is the neutron source term. The flux is analogous to the neutron intensity of the unidirectional neutron beam. There is a need to formulate a relation between neutron flux and neutron current density. Fick’s law provides a convenient mathematical tool. The neutrons diffusion in the reactor is comparable to the motion of solute in the solution (which is originally chemical concept). The current density J (r, t) was found to be proportional to the negative gradient of the neutron flux φ(r, t) according to Fick’s law, i.e J (r, t) = −D∇φ (r, t) .

(7)

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As a matter of fact, Fick’s law is restrictive and based on many assumptions. In spite of all these assumptions, Fick’s law is still applicable in neutron diffusion calculations. For a homogeneous system (after applying Fick’s law), the Eq. (6) can now be written as follows: 1 ∂φ (r, t) = s (r, t) − a (r) φ (r, t) + D∇ 2 φ (r, t) . v ∂t

(8)

The criticality concept in the nuclear reactor theory must be observed carefully. The criticality means that the nuclear reactor must be always under control; this control can be achieved by keeping the number of neutrons constant in every neutron generation. This criticality can be expressed mathematically by keeping the multiplication factor (K ) equals one. In this case, the number of neutrons in the reactor must be constant at all the time. The criticality can make neutron diffusion equation Eq. (8) time-independent equation and can be written as s (r) − a (r) φ (r) + D∇ 2 φ (r) = 0,

(9)

where the source term is given by s(r ) = v f φ, and  f is the fission macroscopic cross sections, we can understand the cross section meaning by considering a beam of neutrons incident on a thin sheet of material, there is a probability to react the neutrons with the material if they get “close enough” to material nucleus. In general, the microscopic cross section (σ) represents the probability of the reaction to take place (the relation between microscopic cross section σ and the macroscopic cross section  is  = σ N a. Now, the time independent neutron diffusion equation Eq. (9) in the core part can be written as follows: (10) ∇ 2 φ (r ) + B 2 φ (r ) = 0. Equation (10) can be obtained from Eq. (9) by substituting the nuclear buckling ν −  + which is important nuclear reactor concept and defined as B 2 = f (D f γ ) . The neutron source s (r) = 0 in the reflected part because there is no any neutron created in this part of the reactor, then the diffusion equation Eq. (9) in this part will be represented as follows: 1 φ (r ) − ∇ 2 φ (r ) = 0, L2

(11)

where L 2 = Da is the diffusion area and D is known as the diffusion coefficient. Now, the first boundary condition is in the reactor centre where the fluxes must have their maximum value (in both spherical and slab reactors) and the neutron current density will be zero, these fluxes will be normalised in numerical examples and the mathematical representation of this condition is

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φc (0) = I, Jc (0) = 0.

(12)

Another position must be considered is in the interface between the core part and the reflector part (r = R), the values of the fluxes φc (x) and φ R (x) are equal, also the neutron currents densities will be equal, and the mathematical representation is φc (R) = φ R (R) , Jc (R) = J R (R) .

(13)

The final boundary condition is in the reactor end, where the flux must be zero at the end of reflector part, we will apply this condition by considering the reflected slab with finite reflected part and reflected sphere with infinite reflected part, which can be represented respectively as follows: φ(a + b) = 0, φ(∞) = 0.

(14)

Both reactor geometries will be considered separately, we will start with a nonsingular problem in slab reactor, after that we will study singular problem arising in the sphere reactor. These nuclear reactor principles which considered above are discussed in details in essential nuclear reactor books [24–27].

3.1 Reflected Slab Reactor Here, we will apply MDTM in the time-independent diffusion equation in the core part of slab reactor (15) ∇ 2 φc (r ) + B 2 φc (r ) = 0, where Bc2 is the bucking in the core part of the slab reactor which considered in Eq. (10). By applying the Laplacian in Cartesian coordinates on the slab reactor, which is one-dimensional reactor, it is straightforward to simplify Eq. (15) as the following: d 2 φc (x) + Bc2 φc (x) = 0. dx2

(16)

Now, we will apply the MDTM in Eq. (16) then the recurrence relation can be written as follows: (k + 1)(k + 2) c (k + 2) = −c (k),

c (0) = Islab , c (1) = 0,

(17)

where Islab is a constant representing the flux value at the centre of the reactor core, and c (k) is the transformed functions of φc (x).

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By solving the recurrence relation in Eq. (17), the series solution is

1 4 4 φc (x) = Islab 1 − 21 Bc2 x 2 + 24 Bc x − ∞  (−1)m (Bc x)2m . = Islab 2m!

1 B6x 6 720 c

+

1 B8x 8 40320 c

+ ···

 (18)

m=0

Thus, the solution of Eq. (16) in the final form is given by φc (x) = Islab cos(Bc x).

(19)

After that, reflected part of the slab reactor will be considered and the neutron diffusion equation is 1 φ R (r ) − ∇ 2 φ R (r ) = 0, (20) L2 where L 2 the diffusion area which considered in Eq. (11). Equation (20) can by simplified by using Laplacian in Cartesian coordinates in one dimension 1 d 2 φ R (y) − 2 φ R (y) = 0, (21) 2 dy LR where x = Dab − y, Dab = a + b and a, b are the dimensions of the core part and reflector part, respectively, in each reactor side. Applying MDTM to Eq. (21) about y = 0 results, the following recurrence relation (k + 1)(k + 2)  R (k + 2) =

1  R (k) ,  R (0) = 0 ,  R (1) = Dslab , L 2R

(22)

where Dslab is a constant representing the flux value at the outer surface of the reflected part of slab reactor. Solving the recurrence relation Eq. (22) results in a series solution given by φ R (x) = Dslab y + 16 L12 y 3 + R ∞  (y/L R )2m+1 . = Dslab (2m+1)!

1 1 120 L 4R

y5 +

1 1 5040 L 6R

y7 +

1 1 362880 L 8R

y9 + · · ·

(23)

m=0

Thus, the final solution of Eq. (20) is given by  φ R (x) = Dslab sinh

Dab − x LR

 .

(24)

This method can reproduce the HPM results [10] for both core part Eq. (19) and reflected part Eq. (24) of the slab reactor with simpler methodology which is easy to deal with.

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By using the boundary conditions in Eq. (13), we have  Islab cos(Bc a) = Dslab sinh

Dab − a LR

DR Islab Bc Dc sin(Bc a) = −Dslab cosh LR



 ,

(25)

R − x0 LR

 ,

(26)

and from Eqs. (25) and (26), we have Bc Dc tan(Bc a) = −

DR coth LR



Dab − a LR

 .

(27)

That can be solved computationally to find the value of a.

3.2 Reflected Spherical Reactor Now, we will consider the spherical reactor and the neutron diffusion equation of its core part (28) ∇ 2 φc (r ) + B 2 φc (r ) = 0. This equation can be written by the definition of Laplacian in spherical coordinates as follows:   ∂ 1 ∂ ∂ 1 1 ∂ 2 ∂ r + sinθ + φ (r, θ, ϑ) + B2 φ (r, θ, ϑ). (29) r 2 ∂r ∂r r 2 sin θ ∂θ ∂θ r 2 sin2 θ ∂ϑ This equation can be simplified to 1 ∂ r2 ∂r

2 ∂ r ∂r φ (r,θ, ϑ) +

1 ∂ r2 sin θ ∂θ

∂ sinθ ∂θ φ (r,θ, ϑ) +

∂ 1 φ (r,θ, ϑ) r2 sin2θ ∂ϑ

+ B2 φ (r,θ, ϑ) = 0.

(30)

At this point, we will use the separation of variables technique by assuming that φ (r, θ, ϑ) = R (r) Y(θ, ϑ) and substituting in the Eq. (30) 1 ∂ r2 ∂r

2 ∂ r ∂r R (r) Y (θ, ϑ) +

∂ 1 r2 sin θ ∂θ

∂ sin sinθ ∂θ R (r) Y (θ, ϑ)

∂ 1 2 + r2 sin 2 ∂ϑ R (r) Y (θ, ϑ) + B R (r) Y (θ, ϑ) = 0. θ

Because R (r ) is (θ, ϑ) independent and Y (θ, ϑ) is r independent

(31)

Analytical Solution of Neutron Diffusion Equation …

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2 ∂ ∂ 1 ∂ ∂ r ∂r R (r) + R (r) r2 sin Y (θ, ϑ) r12 ∂r sin sinθ ∂θ + θ ∂θ

1 r2 sin2θ

∂ ∂ϑ



Y (θ, ϑ)

(32)

+B2 R (r) Y (θ, ϑ) = 0. Divided by R (r ) Y (θ, ϑ), we have 1 1 ∂ R(r) r2 ∂r

2 ∂ r ∂r R (r) +

1 [ 1 ∂ Y(θ,ϑ) r2 sinθ ∂θ

∂ sin sinθ ∂θ

(33)

∂ 1 2 + r2 sin 2 ∂ϑ ]Y (θ, ϑ) + B = 0. θ

Now, our problem is r dependent only, so the second term will be zero and Eq. (33) will be simplified to 1 ∂ 2 ∂ r R (r) + B2 R (r) = 0. r 2 ∂r ∂r

(34)

By considering x = Br , and because the symbol φ is usually used to express the flux [24–27], so Eq. (34) will be x2

d 2 φc (x) dφc (x) + x 2 φc (x) = 0. + 2x 2 dx dx

(35)

Applying MDTM to Eq. (35) results, the following recurrence relation

2  k + k c (k) = −c (k − 2),

c (0) = Ispher e , c (1) = 0,

(36)

where Ispher e is the constant representing the flux value at the centre of the reactor core. Solving this recurrence relation, the series solution of Eq. (36) will be

φc (x) = Ispher e 1 − 16 x 2 + = Ispher e

∞  m=0

1 4 x 120

−

1 x6 5040

+

1 x8 362880

(−1)m (x)2m . (2m+1)!

+ ···

 (37)

Setting x = Bc r in Eq. (37) results in φc (r ) = Ispher e

sin(Bc r ) sin(Bc r ) = Aspher e . Bc r r

(38)

In this stage, the reflected part of spherical reactor will be studied, where the neutron diffusion equation is known as ∇ 2 φ R (r ) −

1 φ R (r ) = 0. L2

(39)

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In the same methodology that used in the core part by applying Laplacian in spherical coordinates, Eq. (39) will be x2

d 2 φ R (x) dφ R (x) + 2x − x 2 φ R (x) = 0. 2 dx dx

(40)

Let x = Dab − y then Eq. (40) is transformed into (Dab − y)2

d 2 φ R (y) dφ R (y) − (Dab − y)2 φ R (y) = 0. + 2(Dab − y) 2 dy dy

(41)

Now, we will use the same procedure that applied in the reflected reactor part of the slab reactor, the solution in its final form is given by  φ R (x) = Dspher e

 Dab cosh(Dab − x) − sinh(Dab − x) , x

(42)

where Dspher e is a constant representing the flux value at the outer surface of the reflector part. Setting x = r/L R and taking in consideration that the flux at infinity should not be infinite, so Eq. (42)

Dab −r/L R +0.5e Dab −r/L R φ R (r ) = L R Dspher e 0.5Dab e r

−r/L Dab −r/L R R = Bspher e e r , ≈ L R Dspher e Dab e r where Bspher e is a constant. To determine a Aspher e sin(Bc a) = Bspher e e−a/L R ,

 Aspher e Dc a Bc cos(Bc a) − sin(Bc a) = −Bspher e D R



(43)

(44)  a + 1 e−a/L R . LR

(45)

Solving Eqs. (44) and (45) results DR a Bc cot(Bc a) = 1 − Dc



 a +1 . LR

(46)

The MDTM results give the same results of HPM [10] for both core part Eq. (37) and reflected part Eq. (43) of the spherical reactor. At the end of this section, after applying MDTM in both slab and spherical reactors, we confirm that MDTM does not need for discretisation or perturbations nor integrations as HPM.

Analytical Solution of Neutron Diffusion Equation …

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4 Results and Comparison The assurance of our study is in verifying the theory and comparing results of MDTM with HPM [10], Cassell and Williams [14] and the transport theory data [28]. We will consider in this section, the use of C++ codes which designed in this work to carry out this target and the origin pro 8 software is used in analyzing the C++ codes data and plot figures. It is important to define diffusion coefficient (D), buckling (B) and diffusion length (L) mathematically as the following: 

1 , D=

3 f +s +γ



νf − f +γ , B= D

 L=

DL , (f +γ )

(47)

where  f , s and γ are the fission, scattering and gamma macroscopic cross sections, and these cross sections are important in representing the probability of occurring of the nuclear reaction in the reactor, these cross sections are constants for each material.

4.1 Reflected Slab Reactor Now, the reflected slab reactor with two cases is considered, the neutron diffusion equation has been solved using MDTM, this solution will be used in achieving numerical results for this reactor. In the first example, the nuclear cross sections when neutrons diffuse in U-D2 O in the core part and the water used in the reflected part where the important reflected slab data is shown in Table 2 [18]. Two C++ codes are written and used, the first is used calculating the critical dimension ac of U-D2 O in the core and water in the reflected part of the slab, the other is used in finding the flux distribution in the reactor. The critical dimension of the reflected slab reactor are compared here with both HPM results and transport theory data as shown in Table 3, where the MDTM can

Table 2 The reflected slab data The core cross sections ν f 1.70 0.054628 The reflector cross sections ν f 0.0000 0.0000

s 0.464338

γ 0.027314

t  f + s + γ

s 0.491652

γ 0.054628

t  f + s + γ

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Table 3 Critical dimension for U-D2 O with H2 O reflector for reflected slab reactor MDTM HPM Transport theory U-D2 O cr

8.6279

8.6279

8.4281

Fig. 1 Flux distribution in reflected slab reactor consists of U-D2 O in the Core, where H2 O is used as a reflector

reproduce the results of HPM and it has fair results when it is compared with the transport theory data. The description of the neutron distribution in the core and reflected parts of the reactor is obtained; this distribution can be expressed by the flux as shown in the Fig. 1. The values of the flux are compared with HPM results only because the data of transport theory are not found, where the flux behaviour can be successfully reproduced. The other case is studied when the reflector thickness is zero; here, the reactor is named as bare reactor, where 1 MeV neutrons diffusing in pure 235 U are considered, the cross section data is given in Table 4 [15, 16]. The critical dimension of the bare slab reactor as shown in Table 5 (where both HPM results and transport theory data are not found), where the MDTM can reproduce the results of HPM and it has fair results when it is compared with the transport theory data. Here, the benchmark data is not available to compare with, but the same methodology used in the reflected slab reactor assures the results. The flux distribution is described in Fig. 2.

Analytical Solution of Neutron Diffusion Equation …

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Table 4 The bare slab data The core cross sections ν σf 2.42 1.336 Na = 0.0478 × 1024 atoms cm−3

σs 5.959

σγ 0.153

Table 5 Critical dimension for 235U for bare slab reactor MDTM HPM Ucr

5.281

–

σt σ f + σs + σγ

Cassell and Williams [14] –

Fig. 2 Flux distribution in 235 U bare slab reactor

As expected, the flux value decreases and reaches its zero value at the reactor critical dimension.

4.2 Reflected Spherical Reactor Finally, we will study the reflected spherical reactor in two cases; more C++ codes are written and used to find the critical dimension and the flux distribution in the spherical reactor.

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The solution of study state diffusion equation is obtained to study this numerical example, 235 U is the core and water is the reflector of this reactor. More needed cross sections are given in Table 6 [11]. The critical radius of the spherical reactor is compared here with both HPM results and transport theory data as shown in Table 7, where the MDTM can reproduce the results of HPM and both methods have fair results when compared with the benchmark (Transport Theory data).

Table 6 The reflected sphere data The core cross sections ν f 2.7971 0.06528 The reflector cross sections ν f 0.0000 0.0000

s 0.24806

γ 0.01306

t  f + s + γ

s 0.24937

γ 0.03264

t  f + s + γ

Table 7 Critical dimension of 235 U with H2 O reflector for reflected spherical reactor MDTM HPM Transport theory Ucr

6.5128

6.5128

Fig. 3 Neutron distributions in spherical reactor of reflector

235 U

6.1275

in the core where water is used as a

Analytical Solution of Neutron Diffusion Equation …

143

The MDTM results have been compared only with HPM because no transport theory data are found. The expected behaviour is shown and assures after this compression in Fig. 3. Another numerical example is studied when the reflector thickness is zero; here, the reactor is bare spherical reactor, where 1 MeV neutrons diffusing in pure 235 U are obtained, the cross section data is given in Table 8 [14, 15]. Here, MDTM reproduces the results in HPM and Cassel and Williams (2004) benchmarks data in good agreement. The critical dimension is given in Table 9 and the flux distribution is described in Fig. 4. As expected the flux values decrease and vanish at the reactor boundary.

Table 8 The bare sphere reactor data The core cross sections ν σf 2.42 1.336 Na = 0.0478 × 1024 atoms cm−3

σs 5.959

σγ 0.153

Table 9 Critical dimension for 235 U for bare slab reactor MDTM HPM Ucr

10.528

10.528

Fig. 4 Flux distribution in 235 U bare sphere reactor

σt σ f + σs + σγ

Cassell and Williams [14] 10.528

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5 Conclusions In this paper, we have presented analytical solutions of time-independent neutron diffusion equations for reflected reactor using MDTM. The MDTM is applied successfully on singular and non-singular initial value problems arising for the slab and spherical geometries. Two computational examples for both reactor geometries are presented and the results are compared with the literature results of HPM, Cassell and Williams [14] and transport theory data. The results assure that the MDTM is effective and reliable in solving the considered problems. Acknowledgements The authors would like to thank the referees for their valuable comments and suggestions, which helped to improve the manuscript. Moreover, the authors thank Prince Sattam bin Abdulaziz University and Deanship of Scientific Research at Prince Sattam bin Abdulaziz University for their continuous support and encouragement.

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14. Cassell, J.S., Williams, M.M.R.: A solution of the neutron diffusion equation for a hemisphere with mixed boundary conditions. Ann. Nucl. Energy 31, 1987–2004 (2004) 15. Khasawneh, K., Dababneh, S., Odibat, Z.: A solution of the neutron diffusion equation in hemispherical symmetry using the homotopy perturbation method. Ann. Nucl. Energy 36, 1711–1717 (2009) 16. Dababneh, S., Khasawneh, K., Odibat, Z.: An alternative solution of the neutron diffusion equation in cylindrical symmetry. Ann. Nucl. Energy 38, 1140–1143 (2010) 17. Shqair, M., El-Ajou, A., Nairat, M.: Analytical solution for multi-energy groups of neutron diffusion equations by a residual power series method. Mathematics 7 (2019) 18. Shqair, M.: Developing a new approaching technique of homotopy perturbation method to solve two-group reflected cylindrical reactor. Results Phys. 12, 1880–1887 (2019) 19. Seiyed, E., Ghasemi, M., Hatami, D., Ganji, D.: Thermal analysis of convective fin with temperature-dependent thermal conductivity and heat generation. Case Stud. Therm. Eng. 4, 1–8 (2014) 20. Nairat, M., Shqair, M., Alhalholy, T.: Cylindrically symmetric fractional Helmholtz equation. Appl. Math. E-Notes 19, 708–717 (2019) 21. Zeidan, D., Strom, H.: Evaluation of a hyperbolic conservative mixture model for thermalhydraulics two-phase flows of nuclear reactors. In: International Conference on Mathematics & Computational Methods Applied to Nuclear Science & Engineering, Jeju, Korea, April 16–20, 2017 (2017) 22. Zeidan, D.: The Riemann problem for a hyperbolic model of two-phase flow in conservative form. Int. J. Comput. Fluid Dyn. 25, 299–318 (2011) 23. Zeidan, D.: Assessment of mixture two-phase flow equations for volcanic flows using Godunovtype methods. Appl. Math. Comput. 272, 707–719 (2016) 24. Duderstadt, J.J., Hamilton, L.J.: Nuclear Reactor Analysis. Wiley, Michigan (1976) 25. Krane, K.S.: Introductory Nuclear Physics. Wiley, New York (1988) 26. Stacey, W.M.: Nuclear Reactor Physics. Wiley-VCH, Weinheim (2018) 27. Lamarsh, J.R.: Introduction to Nuclear Engineering, 2nd ed. Addison-Wesley, Boston (1983) 28. Sood, A., Forster, R., Parsons, D.: Analytical Benchmark Test Set for Criticality Code Verification. Los Alamos National Laboratory, US (2009)

Second-Order Perturbed State-Dependent Sweeping Process with Subsmooth Sets Doria Affane and Mustapha Fateh Yarou

Abstract Using a discretisation approach, the existence of solutions for a class of second-order differential inclusion is stated. The right-hand side of the problem is governed by the so-called nonconvex state-dependent sweeping process and contains an unbounded perturbation, that is the external forces applied on the system. Thanks to some recent concepts of set’s regularity and nonsmooth analysis, we extend existence results for nonconvex equi-uniformly subsmooth sets. The construction is based on Moreau’s catching-up algorithm. Moreover, we extend our result to the more general delayed case, namely when the perturbation contains a finite delay. An example is given for the special case of quasi-variational inequalities which constitutes a variational formulation of certain linear elasticity problems with friction or unilateral constraints.

1 Introduction The perturbed second-order state-dependent nonconvex sweeping process is an evolution differential inclusion governed by the normal cone to a mobile set depending on both time and state variables, of the following form:

D. Affane · M. F. Yarou (B) LMPA Laboratory, Department of Mathematics, Jijel University, Jijel, Algeria e-mail: [email protected] D. Affane e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 D. Zeidan et al. (eds.), Computational Mathematics and Applications, Forum for Interdisciplinary Mathematics, https://doi.org/10.1007/978-981-15-8498-5_7

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(P)

⎧ −u(t) ˙ ∈ N Q(t,v(t)) (u(t)) + F(t, u(t), v(t)), a.e. t ∈ [0, T]; ⎪ ⎪ ⎪ ⎪ t ⎪ ⎪ ⎪ ⎪ v(t) = b + u(s)ds, ∀t ∈ [0, T ]; ⎪ ⎪ ⎨ 0

t ⎪ ⎪ ⎪ ⎪ ˙ ∀t ∈ [0, T ]; u(t) = a + u(s)ds, ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎩ u(t) ∈ Q(t, v(t)), ∀t ∈ [0, T ],

where N Q(t,v(t)) (u(t)) denotes the normal cone of Q(t, v(t)) at the point u(t), the sets Q(t, v(t)) are nonconvex in H and F : [0, T ] × H × H  H is an upper semicontinuous convex valued mapping playing the role of a perturbation to the problem, that is an external force applied on the system. These kinds of problems were initiated by J.J. Moreau (see [22, 23]) for timedependent sets Q(t) and F ≡ 0 to deal with problems arising in elastoplasticity, quasistatics, electrical circuits, hysteresis and dynamics. Since then, various generalisations have been obtained, see for instance, [3, 4, 9–11, 13, 14, 26, 28, 29] and the references therein. When the moving set Q depends also on the state, one obtains a generalisation of the classical sweeping process known as the state-dependent sweeping process. Such problems are motivated by parabolic quasi-variational inequalities arising, e.g. in the evolution of sandpiles, and occur also in the treatment of 2 − D or 3 − D quasistatic evolution problems with friction, as well as in micro-mechanical damage models for iron materials with memory to describe the evolution of the plastic strain in the presence of small damages. We refer to [20] for more details. By means of a generalised version of the Shauder’s fixed point theorem, Castaing, Ibrahim and Yarou [14] provided an approach to prove the existence of solution to (P). The approach is based on Moreau’s catching-up algorithm. For recent results in the study of state-dependent sweeping process, we refer to [1, 2, 19, 25]. Our aim in this paper is two-fold: using some recent concepts of set’s regularity, we show how the approach from [14] can be adapted to yield the existence of solution for (P) with the general class of equi-uniformly subsmooth sets Q(t, x). Moreover, we weaken the usual assumptions on the perturbation by taking F unnecessarily bounded and without any compactness conditions. This result can be extended to the delayed perturbed second-order sweeping process via a reduction approach. The paper is organised as follows: in Sect. 2, we introduce notation and concepts of setvalued and nonsmooth analysis which will be used throughout the paper. In Sect. 3, we prove the main existence result, which will also be used in Sect. 4 to obtain a generalisation to the delayed case via a reduction approach. In the last section, an example is given to show how to apply our results to solve an elasticity problem.

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2 Notations and Preliminaries Throughout the paper, H is a real separable Hilbert space with scalar product denoted ·, · and the associate norm  · . We denote by B the unit closed ball of H and B(a, r ) (resp., B(a, r )) will be the open (resp. closed) ball of centre a ∈ H and radius r > 0, C H ([0, T ]) Banach space of all continuous mappings u : [0, T ] → H endowed with the norm of uniform convergence. For a nonempty closed subset A of H, we denote by d(·, A) the usual distance function associated with A, i.e. d(u, A) = inf u − y, Proj A (u) the projection of u y∈A

onto A defined by Proj A (u) = {y ∈ A : d(u, A) = u − y}. We denote by co(A) the closed convex hull of A, characterised by co(A) = {x ∈ H : ∀x ∈ H, x , x ≤ δ ∗ (x , A)}, where δ ∗ (x , A) = supx , y stands for the support function of A at x ∈ H. Recall y∈A

that for a closed convex subset A, we have   d(x, A) = sup x , x − δ ∗ (x , A) . x ∈B

A subset A is said to be relatively ball compact if for any closed ball B(a, r ) of H, the set B(a, r ) A is relatively compact. A set-valued mapping F : E  H from a Hausdorff topological space E into H is said to be upper semicontinuous if, for any open subset V ⊂ H , the set {x ∈ E : F(x) ⊂ V} is open in E. If ϕ is a real-valued locally Lipschitz function defined on H, Clarke subdifferential ∂ C ϕ(x) of ϕ at x is the nonempty convex compact subset of H, given by ∂ C ϕ(x) = {ξ ∈ H : ϕ◦ (x; v) ≥ ξ, v, ∀v ∈ H }, where ϕ◦ (x; v) = lim sup y→x, t↓0

ϕ(y + tv) − ϕ(y) t

is the generalised directional derivative of ϕ at x in the direction v. We recall (see [16]) that Clarke normal cone to S at x is defined by N S (x) = ∂ C Ψ S (x), where Ψ S denotes the indicator function of S, i.e. Ψ S (x) = 0 if x ∈ S and +∞ otherwise. Another important concept of Fréchet subdifferential will be also needed. A vector v ∈ H is said to be in the Fréchet subdifferential ∂ F ϕ(x) of ϕ at x (see [24]) provided that for every ε > 0 there exists δ > 0 such that for all y ∈ B(x, δ), we have v, y − x ≤ ϕ(y) − ϕ(x) + εy − x.

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It is known that for all x ∈ S, we have ∂ F ϕ(x) ⊂ ∂ϕC (x), N SF (x) ⊂ N S (x), and ∂ F d(x, S) = N SF (x) ∩ B.

(1)

Another important property is that, whenever y ∈ Proj S (x), one has x − y ∈ N SF (y). Now, we introduce the definition of equi-uniform subsmoothness for a family of sets. It is important to emphasise that this class of sets, introduced by Aussel et al. [5], is an extension of convexity and prox-regularity of a set. In this way, the result concerning the existence of solution of the second-order differential inclusion is more general. We begin with some basic definitions from subsmoothness while referring the reader to [5]. Let S be a closed subset of H . We say that S is subsmooth at x ∈ S, if for every ε > 0, there exists δ > 0 such that ξ1 − ξ2 , x1 − x2  ≥ −εx1 − x2 ,

(2)

whenever x1 , x2 ∈ B(x, δ) ∩ S and ξi ∈ N S (xi ) ∩ B, i = 1, 2. The set S is subsmooth, if it is subsmooth at each point of S. We further say that S is uniformly subsmooth, if for every ε > 0 there exists δ > 0, such that (2) holds for all x1 , x2 ∈ S satisfying x1 − x2  < δ and all ξi ∈ N S (xi ) ∩ B. The following subdifferential regularity of the distance function also holds true for subsmooth sets: Proposition 1 ([30]) Let S be a closed set of a Hilbert space. If S is subsmooth at x ∈ S, then N S (x) = N SF (x) and ∂d C (x, S) = ∂ F d(x, S). Definition 1 Let (S(q))q∈I be a family of closed sets of H with parameter q ∈ I. This family is called equi-uniformly subsmooth, if, for every ε > 0, there exists δ > 0 such that, for each q ∈ I, the inequality (2) holds, for all x1 , x2 ∈ S(q) satisfying x1 − x2  < δ and for all ξi ∈ N S(q) (xi ) B, i = 1, 2. For the proofs of the next proposition, we refer the reader to [18]. Proposition 2 Let {C(t, v) : (t, v) ∈ [0, T ] × H } be a family of nonempty closed sets of H which is equi-uniformly subsmooth and let a real η > 0. Assume that there exist real constants L 1 > 0 and L 2 > 0 such that, for any x, u, v ∈ H and s, t ∈ [0, T ]



 |d x, C(t, u) − d x, C(s, v) | ≤ L 1 |t − s| + L 2 u − v. Then the following assertions hold: (a) For all (s, v, y) ∈ Gph(C), we have η∂d(y, C(s, v)) ⊂ ηB;

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(b) The convex weakly compact valued mapping (t, x, y) → ∂ p d(y, C(t, x)) satisfies the upper semicontinuity property: For any sequence (sn )n in [0, T ] converging to s, any sequence (vn )n converging to v, any sequence (yn )n converging to y ∈ C(s, v) with yn ∈ C(sn , vn ) and any ξ ∈ H , we have



 lim sup σ ξ, η∂d(yn , C(sn , vn )) ≤ σ ξ, η∂d(y, C(s, v)) . n→∞

We will also need the following result, which is a discrete version of Gronwall’s Lemma (see [21]). Lemma 1 Let (αi ), (βi ), (γi ) and (ai ) be sequences of nonnegative real numbers such that ai+1 ≤ αi + βi (a0 + · · · + ai−1 ) + (1 + γi )ai , for i ∈ N . Then,

a j ≤ a0 +

j−1

k=0

 αk

 j−1

(kβk + γk ) , for j ∈ N \ {0}. · exp k=0

3 Main Result The following theorem establishes the existence result of the evolution problem (P). Theorem 1 Let Q : [0, T ] × H  H be a set-valued mapping with nonempty closed values satisfying: (Q1 ) the family {Q(t, x); (t, x) ∈ [0, T ] × H } is equi-uniformly subsmooth; (Q2 ) for any bounded subset A ⊂ H, the set Q([0, T ] × A) is relatively ball compact; (Q3 ) there are real constants Λ1 > 0 and Λ2 > 0 such that for all t, s ∈ [0, T ] and z, xi , yi ∈ H, i = 1, 2, |d(z, Q(t, x1 ) − d(z, Q(s, x2 )| ≤ Λ1 |t − s| + Λ2 x1 − x2 . Let F : [0, T ] × H × H  H be an upper semicontinuous set-valued mapping with nonempty closed convex values such that: (F) for some real κ > 0 and for all (t, x, y) ∈ [0, T ] × H × H Pr oj F(t,x,y) (0) ≤ κ(1 + x + y).

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Then, for every (a, b) ∈ H × H with a ∈ Q(0, b) there exists a Lipschitz continuous solution (u, v) to (P) with u˙ (t) ≤ Θ. where

 Θ = Λ1 + Λ2 Δ + 2κ 1 + b + (T + 1)Δ

and     



Δ = Λ1 + 2κ(1 + v) T + a · exp T Λ2 + 2κ(1 + 2T ) . Proof For each (t, x, y) ∈ [0, T ] × H × H , denoted by m(·, ·, ·) the element of minimal norm of the closed convex set F(t, x, y) of H, that is m(t, x, y) = Proj F(t,x,y) (0). For every n ≥ 1, we consider a partition of [0, T ] by the points tkn = ken , en = Tn , k = 0, 1, . . . , n. u n0

Step 1. Let us define inductively the sequences (u nk )0≤k≤n and (vkn )0≤k≤n . We put = a, v0n = b, the following inclusion hold: 

n n n n n u nk+1 ∈ Proj Q(tk+1 ,vkn ) u k − en m(tk , u k , vk )

(3)

n = vkn + en u nk+1 . vk+1

(4)

and

By the property of the Fréchet normal cone and (3), we obtain n n u nk+1 − u nk + en m(tkn , u nk , vkn ) ∈ −N Q(tk+1 ,vkn ) (u k ).

(5)

This algorithm is well defined. Indeed, for k = 0, by the ball compactness of the set Q(t1n , v0n ), we can take

 u n1 ∈ Proj Q(t1n ,v0n ) u n0 − en m(t0n , u n0 , v0n ) and we write v1n = v0n + en u n1 . Further, using (Q3 ) and (F), we get u n1 − u n0  ≤ u n1 − u n0 + en m(t0n , u n0 , v0n ) + en m(t0n , u n0 , v0n )

 = d u n0 − en m(t0n , u n0 , v0n ), Q(t1n , v0n ) + en m(t0n , u n0 , v0n ) ≤ |d(u n0 , Q(t0n , v0n )) − d(u n0 , Q(t1n , v0n ))| + 2en m(t0n , u n0 , v0n ) ≤ Λ1 en + 2κen (1 + u n0  + v0n ). Assume that u n0 , . . . , u nk as well as v0n , . . . , vkn have been constructed satisfying (3) n , vkn ) is ball compact, we can take and (4). Then, since the set Q(tk+1 n n n n n u nk+1 ∈ Proj Q(tk+1 ,vkn ) (u k − en m(tk , u k , vk )).

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For all 0 ≤ i ≤ k, we have n n − u n  ≤ u i+1 − u in + en m(tin , u in , vin ) + en m(tin , u in , vin ) u i+1

ni  n = d u i − en m(tin , u in , vin ), Q(ti+1 , vin ) + en m(tin , u in , vin ) n n ≤ |d(u in , Q(ti+1 , vin )) − d(u in , Q(tin , vi−1 ))| + 2en m(tin , u in , vin ) n n n ≤ Λ1 en + Λ2 vi − vi−1  + 2en m(ti , u in , vin ).

According to relation (4), we obtain, for all 1 ≤ i ≤ k and 1 ≤ k ≤ n, n  ≤ en u in  vin − vi−1

and n + en u in  = v0n + en (u n0 + u n1 + · · · + u in ) vin  = vi−1

 ≤

v0n 

+ en

u n0 

+

u n1 

+··

 .

+u in 

Then, for all 1 ≤ i ≤ k and 1 ≤ k ≤ n, we have n n u i+1 − u in  ≤ Λ1 en + Λ2 vin − vi−1  + 2en κ(1 + u in  + vin )

≤ Λ1 en + Λ2 en u in  + 2en κ(1 + u in  + v0n  + en

i

(6)

u nm )

m=0

    ≤ Λ1 + 2κ(1 + v0n ) en + Λ2 + 2κ(1 + en ) en u in  + 2en2 κ

i−1

u nm .

m=0

Further, for all 0 ≤ i ≤ k and 0 ≤ k ≤ n, we have n n n − u n0  ≤ u i+1 − u in  + u in − u i−1  + · · · + u n1 − u n0 , u i+1

using (6), we get n  u i+1

 

 n n ≤ Λ1 + 2κ(1 + v0 ) T + u 0 

  i−1 i−1

u nm  + 2en2 κ (i − m)u nm  + Λ2 + 2κ(1 + en ) en m=0

m=0

    i−1 

n n u nm . ≤ Λ1 + 2κ(1 + v0 ) T + u 0  + en Λ2 + 2κ(1 + 2T ) m=0

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Using Lemma 1, we get for 1 ≤ i ≤ k ≤ n, i−1   



  u in  ≤ Λ1 + 2κ(1 + v0n ) T + u n0  · exp en Λ2 + 2κ(1 + 2T ) m=0

   



 n n ≤ Λ1 + 2κ(1 + v0 ) T + u 0  · exp T Λ2 + 2κ(1 + 2T ) = Δ. Then for 1 ≤ i ≤ k ≤ n, u in  ≤ Δ, vin  ≤ v0n  + T Δ = Υ and

n

 − u in  u i+1 ≤ Λ1 + Λ2 Δ + 2κ 1 + v0n  + (T + 1)Δ = Θ. en

(7)

(8)

Step 2. Construction of u n (·) and vn (·). n ], k = 0, 1, . . . , n − 1, we define For any t ∈ [tkn , tk+1 vn (t) = vkn + (t − tk )u nk+1 and u n (t) =

n − t n t − tkn n tk+1 uk + u k+1 . en en

n Thus, for almost all t ∈ [tkn , tk+1 ],

u˙ n (t) =

u nk+1 − u nk en

(9)

by relations (5) and (9), we obtain n n n n n −u˙ n (t) ∈ N Q(tk+1 ,vkn ) (u k+1 ) + m(tk , u k , vk )

(10)

||u˙ n (t)|| ≤ Θ.

(11)

by (8), we get

For each t ∈ [0, T ] and each n ≥ 1, we define the functions

Second-Order Perturbed State-Dependent Sweeping …

 pn (t) =  qn (t) =

tkn

if

n tn−1

if

n tk+1

if

tnn

if

155 n t ∈ [tkn , tk+1 [

t = T, n t ∈ [tkn , tk+1 [

t = T.

Observe that, for all t ∈ [0, T ] lim | pn (t) − t| = lim |qn (t) − t| = 0.

n→∞

n→∞

So, we get by (10) −u˙ n (t) ∈ N Q(qn (t),vn ( pn (t))) (u n (qn (t)) + m( pn (t), u n ( pn (t)), vn ( pn (t))) for a.e. t ∈ [0, T ]. It is obvious that, for all n ≥ 1 and for all t ∈ [0, T ], the following hold:   m( pn (t), u n ( pn (t)), vn ( pn (t))) ≤ κ 1 + v0n  + (T + 1)Δ = Λ (12) u n (qn (t)) ∈ Q(qn (t), vn ( pn (t))); t vn (t) = b +

u n ( pn (s))ds,

(13)

lim pn (t) = lim qn (t) = t.

(14)

0

n→∞

n→∞

For k = 1, . . . , n, it result from (7) and (12) that for a.e. t ∈ [0, T ] u n (qn (t)) ∈ Q(qn (t), vn ( pn (t))) ∩ ΔB, so (u n (qn (t))) ⊂ Q([0, T ] × Υ B) ∩ ΔB. By (Q2 ), we deduce that (u n (qn (t))) is relatively compact and qn (t)

u n (qn (t)) − u n (t) ≤

u(s)ds ˙ ≤ Θ(qn (t) − t) → 0, as n → ∞. t

That is, for all t ∈ [0, T ], (u n (t)) is also relatively compact, and, clearly, (u n (·)) is equicontinuous. Therefore, by Ascoli’s Theorem, (u n (·)) is relatively compact in C H ([0, T ]), and we can extract from it a subsequence, that we do not relabel,

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which converges uniformly to some mapping u(·) ∈ C H ([0, T ]). It is obvious that u(0) = a and that u(·) is absolutely continuous. By the inequality (11), there exists a subsequence (again denote by) (u˙ n (·)) which converges σ(L 1H ([0, T ]), L ∞ H ([0, T ])) in L 1H ([0, T ]) to w with w ≤ Θ a.e. t ∈ [0, T ]. Fixing t ∈ [0, T ] and taking any ξ ∈ H, the above weak convergence in L 1H ([0, T ]) yields 



t

lim ξ, u 0 +



u˙ n (s)ds = ξ, u 0 +

n→∞

0



t

w(s)ds . 0

t This means, for each t ∈ [0, T ], that (u n (t)) weakly converges to u 0 + 0 w(s)ds in H . Since the sequence (u n (t)) converges strongly to u(t) in H , it ensures that u˙ = w. From (13), (14) and the convergence of (u n ), we deduce that (vn ) converges t uniformly to an absolutely continuous function v with v(t) = b + 0 u(s) ds. On the other hand, we have ||m( pn (t), u n ( pn (t)), vn ( pn (t))) ≤ Λ for all n ≥ 0 and for all t ∈ [0, T ], we put (m( pn (·), u n ( pn (·)), vn ( pn (·)))) = ( f n (·)), so ( f n (·)) is bounded, taking a subsequence if necessary, we may conclude that ( f n (·)) converges 1 σ(L 1H ([0, T ]), L ∞ H ([0, T ])) to some mapping f ∈ L H ([0, T ]) with  f (t) ≤ Λ. Step 3. We prove, in this step, that the map u is indeed a solution of problem (P). We have d(u n (t), Q(t, v(t))) ≤ ||u n (t) − u n (qn (t))|| + d(u n (qn (t)), Q(t, v(t))) ≤ ||u n (t) − u n (qn (t))|| + |d(u n (qn (t)), Q(t, v(t))) −d(u n (qn (t)), Q(qn (t), v( pn (t)))| ≤ ||u n (t) − u n (qn (t))|| + λ1 |qn (t) − t| + λ2 ||vn ( pn (t)) − v(t)||. Since lim ||u n (qn (t)) − u n (t)|| = lim ||vn ( pn (t)) − v(t)|| = lim |qn (t) − t| = 0,

n→∞

n→∞

n→∞

and Q(t, v(t)) is closed, by passing to the limit in the preceding inequality, we get u(t) ∈ Q(t, v(t)). Now, we have u˙ n (t) + f n (t) ≤ u˙ n (t) +  f n (t) ≤ Θ + Λ = l, that is, u˙ n (t) + f n (t) ∈ lB H . Since u˙ n (t) + f n (t) ∈ −N Q(qn (t),vn ( pn (t))) (u n ( pn (t))) we get by (1)

Second-Order Perturbed State-Dependent Sweeping …

157

u˙ n (t) + f n (t) ∈ −l∂d(u n ( pn (t)), Q(qn (t), vn ( pn (t)))). Note that (u˙ n + f n , f n ) weakly converges in L 1H ×H ([0, T ]) to (u˙ + f, f ). An application of Mazur’s Theorem to (u˙ n + f n , f n ) provides a sequence (wn , ζn ) with wn ∈ co{u˙ m + f m : m ≥ n} and ζn ∈ co{ f m : m ≥ n} and such that (wn , ζn ) converges strongly in L 1H ×H ([0, T ]) to (u˙ + f, f ). We can extract from (wn , ζn ) a subsequence which converges a.e. to (u˙ + f, f ). Then, there is a Lebesgue negligible set N ⊂ [0, T ] such that for every t ∈ [0, T ] \ N u(t) ˙ + f (t) ∈



{wm (t) : m ≥ n} ⊂

n≥0

and f (t) ∈



co{u˙ m (t) + f m (t) : m ≥ n}

(15)

n≥0



{ζm (t) : m ≥ n} ⊂

n≥0



co{ f m (t) : m ≥ n}.

n≥0

Fix any t ∈ [0, T ] \ N , n ≥ n 0 and μ ∈ H, then the relation (15) gives μ, u(t) ˙ + f (t) ≤ lim supδ ∗ (μ, −l∂d(u n ( pn (t)), Q(qn (t), vn ( pn (t))))). n→∞

By Proposition 2, we obtain μ, u(t) ˙ + f (t) ≤ δ ∗ (μ, −l∂d(u(t), Q(t, v(t)))), which entails u(t) ˙ + f (t) ∈ −l∂d(u(t), Q(t, v(t))) ⊂ −N Q(t,v(t)) (u(t)). Further, the relation (16) gives μ, f (t) ≤ lim sup δ ∗ (μ, F( pn (t), u n ( pn (t)), vn ( pn (t)))). n→∞

Since δ ∗ (μ, F(·, ·, ·)) is upper semicontinuous on [0, T ] × H × H then μ, f (t) ≤ δ ∗ (μ, F(t, u(t), v(t))), so, we get d( f (t), F(t, u(t), v(t))) ≤ 0, consequently f (t) ∈ F(t, u(t), v(t)) a.e. t ∈ [0, T]. This completes the proof of the Theorem.

(16)

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Remark 1 Many authors studied some variants of the sweeping process. In [1], the following implicit sweeping process was considered: 

u(t) ˙ ∈ −N Q(t,u(t)) (Au(t) ˙ + Bu(t)), a.e. t ∈ [0, T]; u(0) = u 0 ,

A similar approach provides a variant of our problem (P) as follows: under the assumptions of Theorem 1, let A : H → H be a bounded linear symmetric operator such that Ax, x > 0 for any β > 0, there exists a solution to

(P )

⎧ −u(t) ˙ ∈ N Q(t,v(t)) (Au(t)) + F(t, u(t), v(t)), a.e. t ∈ [0, T]; ⎪ ⎪ ⎪ ⎪ t ⎪ ⎪ ⎪ ⎪ v(t) = b + u(s)ds, ∀t ∈ [0, T ]; ⎪ ⎪ ⎨ 0

t ⎪ ⎪ ⎪ ⎪ ˙ ∀t ∈ [0, T ]; u(t) = a + u(s)ds, ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎩ u(t) ∈ Q(t, v(t)), ∀t ∈ [0, T ],

4 Delayed Sweeping Process Now, we proceed, in the infinite dimensional setting, to an existence result for secondorder functional differential inclusion governed by the time and state-dependent nonconvex sweeping process, that is when the perturbation contains a finite delay. This problem was addressed by [25] using the discretisation approach based on Moreau’s catching-up algorithm. Here, we provide an other technique initiated in [12] for the first-order time-dependent case, which consists to subdivide the interval [0, T ] in a sequence of subintervals and to reformulate the problem with delay to a sequence of problems without delay and apply the results known in this case. For second-order functional problems regarding the time-dependent sweeping process, we refer to [8, 9]. We will extend this approach for the case of time and statedependent sweeping process with unbounded delayed perturbation. Let τ > 0 be a positive number and C0 = C H ([−τ , 0]) (resp. CT = C H ([−τ , T ]) Banach space of H -valued continuous functions defined on [−τ , 0] (resp. [−τ , T ]) equipped with the norm of uniform convergence. Let u : [−τ , T ] → H , then for every t ∈ [0, T ], we define the function u t = T (t)u on [−τ , 0] by (T (t)u) (s) = u (t + s) , ∀s ∈ [−τ , 0]. Clearly, if u ∈ CT , then u t ∈ C0 and the mapping u → u t is continuous. Consider the following problem

Second-Order Perturbed State-Dependent Sweeping …

159

  ⎧ ⎪

 ⎪ − u(t) ˙ ∈ N (u(t)) + G t, T (t)v, T (t)u a.e. t ∈ [0, T ]; ⎪ ⎪ Q t,v(t) ⎪ ⎪ ⎪ ⎪ t t ⎨ u(t) = ψ(0) + v(s)ds, ˙ v(t) = ϕ(0) + u(s)ds, ∀t ∈ [0, T ]; (Pτ ) ⎪ ⎪ ⎪ 0 0 ⎪

⎪ ⎪ ⎪ ⎪ v(t) ∈ Q t, u(t) , ∀t ∈ [0, T ]; ⎩ u ≡ ψ and v ≡ ϕ on [−τ , 0]. Theorem 2 Assume that Q : [0, T ] × H  H satisfies assumptions of Theorem 1 and let G : [0, T ] × C0 × C0  H be a set-valued mapping with nonempty closed convex values such that: (G1 ) (G2 )

G is the upper semicontinuous on [0, T ] × C0 × C0 ; there exists a real β > 0, such that, for all (t, ϕ, ψ) ∈ [T0 , T ] × C0 × C0 , d(0, G(t, ϕ, ψ)) ≤ κ(1 + ϕ(0) + ψ(0)).

Then for every (ϕ, ψ) ∈ C0 × C0 verifying ψ (0) ∈ Q (0, ϕ (0)), there exist two absolutely continuous mappings u : [0, T ] → H and v : [0, T ] → H solution of (Pτ ). Proof Let a = ψ (0) and b = ϕ (0) , then a ∈ Q (0, b) . We consider the same partition of [0, T ] by the points tkn = ken, en = Tn , (k = 0, 1, . . . , n) . For each (t, u, v) ∈       −τ , t1n × H × H, we define f 0n : −τ , t1n × H → H, g0n : −τ , t1n × H → H by  ϕ (t) ∀t ∈ [−τ , 0] , n f 0 (t, v) = ϕ (0) + Tn t (v − ϕ (0)) ∀t ∈ ] 0, t1n ] ,  g0n

(t, u) =

ψ (t) ∀t ∈ [−τ , 0] , ψ (0) + Tn t (u − ψ (0)) ∀t ∈ ] 0, t1n ] .



  v and g0n t1n , v = u for all (u,v) ∈ H × H. Observe that We have f 0n t1n , v =  the mapping (u, v) → T (t1n ) f 0n (·, v) , T (t1n )g0n (·, u) from H × H to C0 × C0 is nonexpansive since, for all (v1 , v2 ) ∈ H × H ,   T (t n ) f n (·, v1 ) − T (t n ) f n (·, v2 ) 1 0 1 0 C 



0  = sup  f 0n s + t1n , v1 − f 0n s + t1n , v2  s∈[−τ ,0]   n  f (s, v1 ) − f n (s, v2 ) = sup 0 0 s∈[−τ + Tn , Tn ] n  n   = sup  s (v1 − ϕ (0)) − s (v2 − ϕ (0)) T T T 0≤s≤ n n    = sup  s (v1 − v2 ) = v1 − v2  . T T 0≤s≤ n

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Similarly, for all (u 1 , u 2 ) ∈ H × H , we get   T (t n )g n (·, u 1 ) − T (t n )g n (·, u 2 ) 1

0

1

0

C0

= u 1 − u 2  .

  Hence, the mapping (u, v) → T (t1n ) f 0n (·, v) , T (t1n )g0n (·, u) from H × H to so the set-valued mapping with nonempty closed convex C0 × C0 is nonexpansive,   values G n0 : 0, t1n × H × H  H defined by G n0 (t, v, u) = G(t, T (t1n ) f 0n (·, v) , T (t1n )g0n (·, u)) is upper semicontinuous thanks to (G1 ) and d(0, G n0 (t, v, u)) = d(0, G(t, T (t1n ) f 0n (·, v) , T (t1n )g0n (·, u))) ≤ κ (1 + v + u) ,   for all (t, v, u) ∈ 0, t1n × H × H since, T (t1n ) f 0n (0, v) = v, T (t1n )g0n (0, u) = u. Hence G n0 verifies conditions  of Theorem 1,   then there exist two absolutely continuous mappings u n0 : 0, t1n → H and v0n : 0, t1n → H such that

   ⎧ n −u˙ 0 (t) ∈ N Q(t,u n0 (t)) v0n (t) + G n0 (t, v0n , u n0 ) a.e on 0, t1n ; ⎪ ⎪ ⎪ ⎪ t t ⎪ ⎪   ⎨ n n n v0 (t) = b + u 0 (s) ds, u 0 (t) = a + u˙ n0 (s) ds ∀t ∈ 0, t1n ; ⎪ 0 0 ⎪  n ⎪ n n ⎪ u ; (t) ∈ Q(t, v ∀t ∈ 0, t (t)) ⎪ 0 1 ⎪ 0 ⎩ v0n (0) = b = ϕ (0) , u n0 (0) = a = ψ (0) , with

 n  u˙ (t) ≤ Θ. 0



Set vn (t) =

 u n (t) =

ϕ (t) ∀t ∈ [−τ , 0] , v0n (t) ∀t ∈ ] 0, t1n ] . ψ (t) ∀t ∈ [−τ , 0] , u n0 (t) ∀t ∈ ] 0, t1n ] .

  Then, u n and vn are well defined on −τ , t1n , with vn = ϕ, u n = ψ on [−τ , 0], and

Second-Order Perturbed State-Dependent Sweeping …

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⎧  n ⎪ ⎪ −u˙ n (t) ∈ N Q(t,vn (t)) (u n (t)) + G 0 (t, vn (t), u n (t)) a.e on 0, t1 ; ⎪ ⎪ ⎪ t ⎪ ⎪ ⎪ ⎪ ⎪ vn (t) = b + u n (s) ds, ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎨ t   ⎪ ⎪ u n (t) = a + u˙ n (s) ds, ∀t ∈ 0, t1n ; ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪  n ⎪ ⎪ ⎪ u (t) ∈ Q(t, v n (t)), ∀t ∈ 0, t1 ; ⎪ n ⎪ ⎪ ⎩ vn (0) = b = ϕ (0) , u n (0) = a = ψ (0) .   By induction, suppose that u n and vn are defined on −τ , tkn (k ≥ 1) with vn = ϕ, u n = ψ on [−τ , 0] and satisfy

vn (t) =

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

t v0n

(t) = b +

  u n (s) ds ∀t ∈ 0, t1n ,

0

v1n

 (t) = vn t1n +

t u n (s) ds ∀t ∈ ] t1n , t2n ] ,

t1n ⎪ ⎪ ⎪ ⎪ ⎪ · · · ⎪ ⎪ ⎪ ⎪ t ⎪  ⎪ ⎪

n  ⎪ n n ⎪ v t + u n (s) ds ∀t ∈ ] tk−1 , tkn ] , = v (t) ⎪ n k−1 k−1 ⎪ ⎪ ⎩ n tk−1

u n (t) =

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

 u n0 (t) = b + u n1

t

0

 (t) = u n t1n +

  u˙ n (s) ds ∀t ∈ 0, t1n ;

t u˙ n (s) ds ∀t ∈ ] t1n , t2n ] ; t1n

⎪ ⎪ ⎪ · · · ⎪ ⎪ ⎪ ⎪ t ⎪  ⎪ ⎪

n  ⎪ n n ⎪ u t + u˙ n (s) ds ∀t ∈ ] tk−1 , tkn ] , = u (t) ⎪ n k−1 ⎪ ⎪ k−1 ⎩ n tk−1

u n and vn are solution of

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  ⎧ ⎪ n n n n ⎪ − u ˙ (t) ∈ N (u (t)) + G t, T (t ) f (·, v (t)), T (t )g (·, u (t)) ; ⎪ n Q(t,vn (t)) n n n k k−1 k k−1 ⎪ ⎪ ⎪ ⎪ ⎪ t ⎪ ⎪

n  ⎪ n ⎪ u n (s) ds; ⎪ ⎨ vn (t) = vk−1 (t) = vn tk−1 + tn

k−1 ⎪ ⎪ t ⎪ ⎪

n  ⎪ n ⎪ u n (t) = u k−1 (t) = u n tk−1 + u˙ n (s) ds; ⎪ ⎪ ⎪ ⎪ ⎪ n ⎪ tk−1 ⎪ ⎩ u n (t) ∈ Q(t, vn (t))

n n n on ] tk−1 , tkn ] , where f k−1 and gk−1 are defined for any (v, u) ∈ H × H as follows

  n , ∀t ∈ −τ , tk−1 vn (t)



 



 n (17) (t, v) = n n n n t − tk−1 + v − vn tk−1 ∀t ∈ ] tk−1 , tkn ] . vn tk−1 T    n , ∀t ∈ −τ , tk−1 u n (t) n

n  n



n  gk−1 (18) (t, u) = n n n t − tk−1 u − u n tk−1 ∀t ∈ ] tk−1 , tk ] . u n tk−1 + T   n Similarly, we can define f kn , gkn : −τ , tk+1 × H → H as 

n f k−1

  ∀t ∈ −τ , tkn , vn (t)



 



 n (t, v) = n t − tkn v − vn tkn ∀t ∈ ] tkn , tk+1 vn tkn + ]; T    u n (t) ∀t ∈ −τ , tkn , n



 



 n gk (t, u) = n t − tkn u − u n tkn ∀t ∈ ] tkn , tk+1 u n tkn + ]; T 

f kn

for any (u, v) ∈ H × H . Note that for all (u, v) ∈ H × H ,

n  n ) f kn (0, v) = f kn tk+1 , v = v, T (tk+1

n  n )gkn (0, u) = gkn tk+1 , u = u. T (tk+1 Note also that, for all (u 1 , v1 ) , (u 2 , v2 ) ∈ H × H, we have   T (t n ) f n (·, v1 ) − T (t n ) f n (·, v2 ) k+1 k k+1 k C 



0n  n = sup  f kn s + tk+1 , v1 − f kn s + tk+1 , v2  s∈[−τ ,0]   n  f (s, v1 ) − f n (s, v2 ) , =  sup k k  s∈ −τ + (k+1)T , (k+1)T n n

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163

and   T (t n )g n (·, u 1 ) − T (t n )g n (·, u 2 ) k+1 k k+1 k C0  n



 n n n  = sup gk s + tk+1 , u 1 − gk s + tk+1 , u2  s∈[−τ ,0]   n g (s, u 1 ) − g n (s, u 2 ) . =  sup k k  s∈ −τ + (k+1)T , (k+1)T n n

We distinguish two cases (1) if −τ +

kT (k + 1) T < , we have n n  n   f (s, v1 ) − f n (s, v2 ) sup k k   s∈ −τ + (k+1)T , (k+1)T n n

=

sup

  n  f (s, v1 ) − f n (s, v2 ) k k

sup

n

    s − tkn (v1 − v2 ) = v1 − v2  ,  T

  (k+1)T s∈ kT n , n

= kT n

≤s≤ (k+1)T n

and sup

  s∈ −τ + (k+1)T , (k+1)T n n

=

kT n

(2) if

kT n

≤ −τ +

sup

n

    s − tkn (u 1 − u 2 ) = u 1 − u 2  .  T

≤s≤ (k+1)T n

≤

k

(k+1)T n

sup

  s∈ −τ + (k+1)T , (k+1)T n n

kT n

and

k

, we have  n   f (s, v1 ) − f n (s, v2 ) k k

sup

 n   f (s, v1 ) − f n (s, v2 ) k k

sup

n     (s − tkn ) (v1 − v2 ) = v1 − v2  T

  (k+1)T s∈ kT n , n

=

k

 n  g (s, u 1 ) − g n (s, u 2 )

(k+1)T n

=

k

sup

  (k+1)T s∈ kT n , n

=

 n  g (s, u 1 ) − g n (s, u 2 )

≤s≤ (k+1)T n

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sup

  s∈ −τ + (k+1)T , (k+1)T n n

=

kT n

k

k

sup

 n  g (s, u 1 ) − g n (s, u 2 )

sup

n     (s − tkn ) (u 1 − u 2 ) = u 1 − u 2  . T

  (k+1)T s∈ kT n , n

=

 n  g (s, u 1 ) − g n (s, u 2 )

≤s≤ (k+1)T n

k

k

  n So the mapping (v, u) → T (tk+1 ) f kn (·, v) , T (tk+1 )gkn (·, u) from H × H to   n C0 × C0 is nonexpansive. Hence the set-valued mapping G nk : tkn , tk+1 ×H×H H defined by   n n ) f kn (·, v) , T (tk+1 )gkn (·, u) G nk (t, v, u) = G t, T (tk+1 is upper semicontinuous with nonempty closed convex values. As above we can easily check that   n d(0, G nk (t, v, u)) ≤ κ (1 + u + v) , ∀ (t, u, v) ∈ tkn , tk+1 × H × H. Applying Theorem 1, there exist   two absolutely continuous mappings   n n → H and vkn : tkn , tk+1 → H such that u nk : tkn , tk+1 ⎧ n

   n −u˙ k (t) ∈ N Q(t,vkn (t)) u nk (t) + G nk (t, vkn (t) , u nk (t)) a.e. on tkn , tk+1 ; ⎪ ⎪ ⎪ ⎪ t ⎪ ⎪

   ⎪ n n ⎪ vk (t) = vn tkn + u nk (s) ds, ∀t ∈ tkn , tk+1 ; ⎪ ⎪ ⎪ ⎨ n tk

t ⎪ ⎪

n n n  ⎪ n n ⎪ ⎪ ⎪ u k (t) = u n tk + u˙ k (s) ds, ∀t ∈ tk , tk+1 ; ⎪ ⎪ ⎪ ⎪ tkn ⎪   ⎩ n n u k (t) ∈ Q(t, vkn (t)) ∀t ∈ tkn , tk+1 , with u˙ nk (t)  ≤ Θ. Thus, by induction, we can construct two continuous mappings u n , vn : [−τ , T ] → H × H with  ϕ (t) ∀t ∈ [−τ , 0] , vn (t) = n vkn (t) ∀t ∈ ] tkn , tk+1 ] , ∀k = 0, . . . , n − 1;  u n (t) =

ψ (t) ∀t ∈ [−τ , 0] , n u nk (t) ∀t ∈ ] tkn , tk+1 ] , ∀k = 0, . . . , n − 1,

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165

  n is a pair solution to such that their restriction on each interval tkn , tk+1 ⎧ n n −u˙ (t) ∈ N Q(t,v(t)) (u(t)) + G(t, T (tk+1 ) f kn (., v (t)) , T (tk+1 )gkn (., u (t))); ⎪ ⎪ ⎪ t t ⎪ ⎨



 v (t) = vn tkn + u (s) ds, u (t) = u n tkn + u˙ (s) ds ⎪ ⎪ ⎪ tkn tkn ⎪ ⎩ u (t) ∈ Q(t, v (t)).   n × C0 × C0 be the element of minimal norm of G nk , then Let h nk : tkn , tk+1 ⎧ n n    n n h t, v (t) , u nk (t) ∈ , vkn (t) , u nk (t)) a.e. on tkn , tk+1 ⎪ k (t, ⎪ 

G   ⎨ k n k n n n n n n n −u˙ k (t) (t) + h , tk+1 , u t, v ∈ N , u a.e. on t (t) (t) Q(t,v (t)) k k k k k k 





 v n t n = vn tkn , u nk tkn = u n tkn  ⎪ ⎪ ⎩ kn k n u k (t) ∈ Q(t, vkn (t)), ∀t ∈ tkn , tk+1 . n Set for notational convenience, h n (t, v, u) = h nk (t, v, u), θn (t) = tk+1 and δn (t) = n n n tk , for all t ∈ ] tk , tk+1 ] . Then we get for almost every t ∈ [0, T ]

⎧ h n (t, vn , u n ) ∈ G(t, T (θn (t)) f nn δn (t) (·, vn (t)) , T (θn (t))g nn δn (t) (·, u n (t))); ⎪ ⎪ T T ⎨ −u˙ n (t) ∈ N Q(t,vn (θn (t))) (u n (θn (t))) + h n (t, vn (t), u n (t)) ; ⎪ v (0) = b = ϕ (0) , u n (0) = a = ψ (0) ∈ Q (0, b) , ⎪ ⎩ n u n (t) ∈ Q (t, vn (θn (t))) , ∀t ∈ [0, T ] with for all t ∈ [0, T ]   d 0, G(t, T (θn (t)) f nn δn (t) (·, vn (t)) , T (θn (t))g nn δn (t) (·, u n (t)) T

T

≤ κ (1 + u n (t) + vn (t)) . We claim that T (θn (t)) f nn δn (t) (·, vn (t)) and T (θn (t))g nn δn (t) (·, u n (t)) pointwise T T converge on [0, T ] to T (t)v and T (t)u, respectively in C0 . The proof is similar to the one given in the Theorem 2.1 in [15]. Further, as u(t) ˙ ≤ Θ , u(t) ≤ a + T Θ and v (t) ≤ b + T (a + T Θ), we get   h n (t, vn (t), u n (t))  ≤ κ 1 + u n (t) + vn (t)

 ≤ κ 1 + a + b + T a + T Θ(1 + T ) . We can proceed as in Theorem 1 to conclude the convergence of (u n ) and (vn ) to the solution of (Pτ ).

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5 Examples 1. In the convex case, Moreau ([23]) introduced second-order Measure differential inclusions, that is an extension of the sweeping process for Lagrangian systems subject to frictionless unilateral constraints and taking the form dv ∈ −NC(u(t)) (v(t)) , v(t) ∈ C(u(t)), where v(·) is the first derivative of u(·) and C(u(t)) is a closed convex set-valued mapping. Numerical time integration schemes have been used for such problems, convergence and well-posedness have been established. This type of problem contains a special case which deserves great attention, namely variational and quasi-variational inequalities of the form find u(t) ∈ C(u(t)) : u(t) ˙ + f (t), v − u(t) ≥ 0 , ∀v ∈ C(u(t)) which is equivalent to −u(t) ˙ ∈ NC(u(t)) (u(t)) + f (t). Let consider the following problem : find u : [0, T ] → H with absolutely continuous derivative u˙ such that for any u 0 = u(0) ∈ C(u(0)), and for any v ∈ C(u(t)) one has u(t) ¨ + g(t), v − u(t) ˙ ≥  f (t, u(t), u(t))), ˙ v − u(t) ˙ ˙ ∈ F(t, u(t), u(t))). ˙ The problem where g ∈ L 1 ([0, T ], H ) and f (t, u(t), u(t))) is equivalent to ˙ + f (t, u(t), u(t))). ˙ u(t) ¨ + g(t) ∈ NC(u(t)) (u(t)) Taking x(t) = u(t) +

t 0

g(s)ds and ⎛

Q(t, x(t)) = C ⎝x(t) −

t 0

⎞ g(s)ds ⎠ +

t g(s)ds, 0

we obtain ˙ + F(t, x(t), x(t)). ˙ −x(t) ¨ ∈ N Q(t,x(t)) (x(t)) This class of problems find its motivation in the fact that it constitutes the variational formulation for linear elasticity problems with friction or unilateralconstraints. For example, let Ω be an open set in R3 with boundary Γ = ΓU ΓF and consider a field of displacement u = u(x, t) satisfying

Second-Order Perturbed State-Dependent Sweeping …

167

∂2ui = σi j, j + f i in Q = Ω×]0, T [ ; ∂t 2 σi j = ai jkh εkh (u), with the boundary conditions u i = Ui on ΓU ×]0, T [= ΣU σ N = FN on ΓF ×]0, T [= ΣF |σT | < g ⇒

∂u T ∂u T = 0 , |σT | = g ⇒ ∃λ ≥ 0 : = −λσT ∂t ∂t

and the initial conditions u(x, 0) = u 0 (x) ,

∂u(x, 0) = u 1 (x) , x ∈ Ω ∂t

where one assumes that the displacements are given on ΓU and that the surface forces are given on ΓF of the boundary, ΓU and ΓF do not depend on time. Ui is the vector field prescribed on ΓU , possibly dependent on time. σi j stands for the stress tensor, εkh (u) the linearised strain tensor and ai jkh the coefficients of elasticity, independent of the strain tensor. We refer to [17] (see also [27]) for the physical interpretation and the following variational formulation of the problem: ˙ ∈ Uad (t) ∀ t ∈ [0, T ], Find u : [0, T ] → R3 such that u(t) where Uad (t) := {v ∈ (H 1 (Ω))3 : v = U˙ (T ) on U } and u(t), ¨ v − u(t) ˙ + a(u(t), v − u(t)) ˙ + ϕ(v) − ϕ(u(t)) ˙ ≥  f (t, u(t)), v − u(t) ˙ for all v ∈ Uad (t)

˙ = u1, u(0) = u 0 , u(0)

where a(·, ·) : H × H → R is a continuous bilinear and symmetric form. See also [1] for a similar problem. Following [1], one proves the equivalence between this variational inequality and the following variant of our perturbed state-dependent sweeping process: −u(t) ˙ ∈ NC(t,v(t)) (Au(t)). 2. More generally, the case when the sets are uniformly prox regular (nonconvex and subsmooth) is appropriate in the study of evolution problem where the state variable is subjected to some constraints and therefore it has to stay in an admissible set. Such second-order differential inclusions appear in several fields: granular

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media, electricity, robotics and virtual reality, see [6] for more details. Another interesting example is studied in [7] concerning the modelling of crowd motion in an emergency evacuation, where the uniform prox-regularity may not be satisfied or may be difficult to be checked. The authors introduced a weaker notion corresponding to a ‘directional prox-regularity property’. The subsmooth case will be the subject of a forthcoming research project.

6 Conclusion In this work, using a discretisation approach, we give a new existence result for the second-order state-dependent sweeping process with the class of equi-uniformly subsmooth sets that are a generalization: sub-smooth sets which are a generalization of convex and uniformly prox-regular sets. The fact that the constraint Q(t, x) depends upon the unknown state x makes the study of the evolution quasi-variational inequalities more complicate. By using the catching-up algorithm, we obtain the strong convergence of approximate functions to the solution of the problem, which is an important property in view of the construction of adequate numerical algorithms. Furthermore, the perturbation isn’t bounded, we require just that the element of minimal norm satisfies a linear growth condition, which is weaker than the ones used in previous works. We generalise this result to the more general delayed case, we show also examples when one can apply these results. Acknowledgements Research supported by the General direction of scientific research and technological development (DGRSDT) under project PRFU No. C00L03UN 180120180001.

References 1. Adly, S., Haddad, T., Le, B.K.: State-dependent implicit sweeping process in the framework of quasistatic evolution quasi-variational inequalities. J. Optim. Theory Appl. 182(2), 473–493 (2019) 2. Affane, D., Yarou, M. F.: Unbounded perturbation for a class of variational inequalities. Discuss. Math. Diff. Inclusion Control Optim. 37, 83–99 (2017) 3. Affane, D., Aissous, M., Yarou, M.F.: Existence results for sweeping process with almost convex perturbation. Bull. Math. Soc. Sci. Math. Roumanie 61(109), 119–134 (2018) 4. Affane, D., Aissous, M., Yarou, M.F.: Almost mixed semicontinuous perturbations of Moreau’s sweeping processes. Evol. Eqs. Contr. Theo. 09(01), 27–38 (2020) 5. Aussel, D., Daniilidis, A., Thibault, L.: Subsmooth sets: function characterizations and related concepts. Trans. Am. Math. Soc. 357, 1275–1301 (2005) 6. Bernicot, F., Venel, J.: Existence of solutions for second-order differential inclusions involving proximal normal cones. J. Math. Pures Appl. 98, 257–294 (2012) 7. Bernicot, F., Venel, J.: Differential inclusions with proximal normal cones in Banach spaces. J. Convex Anal. 17(2), 451–484 (2010) 8. Bounkhel, M., Yarou, M.F.: Existence results for nonconvex sweeping process with perturbation and with delay: Lipschitz case. Arab J. Math. 8(2), 1–12 (2002)

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9. Bounkhel, M., Yarou, M.F.: Existence results for first and second order nonconvex sweeping process with delay. Port. Math. 61(2), 2007–2030 (2004) 10. Brogliato, B.: The absolute stability problem and the Lagrange-Dirichlet theorem with monotone multivalued mappings. Syst. Control Lett. 51(5), 343–353 (2004) 11. Castaing, C.: Quelques problèmes d’évolution du second ordre. Sem. Anal. Convexe, Montpellier, Exposé No 5, (1988) 12. Castaing, C., Ibrahim, A.G.: Functional differential inclusion on closed sets in Banach spaces. Adv. Math. Econ. 2, 21–39 (2000) 13. Castaing, C., Ibrahim, A.G., Yarou, M.F.: Existence problems in second order evolution inclusions: discretization and variational approach. Taiwanese J. Math. 12(6), 1435–1477 (2008) 14. Castaing, C., Ibrahim, A.G., Yarou, M.F.: Some contributions to nonconvex sweeping process. J. Nonlinear Convex Anal. 10, 1–20 (2009) 15. Castaing, C., Salvadori, A., Thibault, L.: Functional evolution equations governed by nonconvex sweeeping process. J. Nonlinear Convex Anal. 2(2), 217–241 (2001) 16. Clarke, F.H., Stern, R.L., Wolenski, P.R.: Proximal smoothness and the lower C 2 property. J. Convex Anal. 02, 117–144 (1995) 17. Duvaut. D., Lions, J. L.: Inequalities in Mechanics and Physics, pp. 1–397. Springer, Berlin (1976) 18. Haddad, T., Noel, J., Thibault, L.: Perturbed sweeping process with a subsmooth set depending on the state. J. Linear Nonlinear Anal. 2(1), 155–174 (2013) 19. Jourani, A., Vilches, E.: Moreau-Yosida regularization of state-dependent sweeping processes with nonregular sets. J. Optim. Theory Appl. 173, 91–116 (2017) 20. Kunze, M., Monteiro Marques, M. D. P.: An Introduction to Moreau’s Sweeping Process. Lectures Notes in Physics, 551, pp. 1–60. Springer, Berlin (2000) 21. Kunze, M., Monteiro Marques, M.D.P.: BV solutions to evolution problems with timedependent domains. Set-Valued Anal. 5, 57–72 (1997) 22. Moreau, J.J.: Evolution problem asssociated with a moving convex set in a Hilbert space. J. Diff. Equat. 26, 347–374 (1977) 23. Moreau, J.J.: Unilateral contact and dry friction in finite freedom dynamics. In: Nonsmooth Mechanics and Applications, 302 CISM, Courses and lectures, pp. 1–82. Springer (1988) 24. Mordukhovich, B.S., Shao, Y.: Nonsmooth sequential analysis in Asplund spaces. Trans. Am. Math. Soc. 4, 1235–1279 (1996) 25. Noel, J.: Second-order general perturbed sweeping process differential inclusion. J. Fixed Point Theory Appl. 20(3), 133, 1–21 (2018) 26. Noel, J., Thibault, L.: Nonconvex sweeping process with a moving set depending on the state. Vietnam J. Math. 42, 595–612 (2014) 27. Sofonea, M., Matei, A.: Variational inequalities with applications. In: Advances in Mechanics and Mathematics, vol. 18, pp. 1–230. Springer, New York (2009) 28. Saïdi, S., Thibault, L., Yarou, M.F.: Relaxation of optimal control problems involving time dependent subdifferential operators. Numer. Funct. Anal. Optim. 34(10), 1156–1186 (2013) 29. Saïdi, S., Yarou, M.F.: Set-valued perturbation for time dependent subdifferential operator. Topol. Meth. Nonlinear Anal. 46(1), 447–470 (2015) 30. Thibault, L.: Subsmooth functions and sets. J. Linear Nonlinear Anal. 4(2), 257–269 (2018)

Membrane Hydrogen Mixture Separation: Modelling and Analysis Khaled Alhussan, Kirill Delendik, Natalia Kolyago, Oleg Penyazkov, and Olga Voitik

1 Introduction Our planet is confronted by global warming due to greenhouse gas emissions. Hydrogen is a clean energy carrier and a hydrogen-based energy sector is regarded as a promising solution for the future of energy security and stability. Though the road to hydrogen energy may be a long, tortuous path, implementing hydrogen production during a period of petroleum instability can facilitate the evolution of the petroleum economy to a hydrogen economy [1]. There are three basic methods for hydrogen production: water electrolysis, steam methane conversion, gasification of coal and biomass [2]. The low-consumption method is steam methane conversion, but additionally, hydrogen needs to be purified up to the ultrahigh grade desired for fuel cells [3]. Currently, the membrane technology for hydrogen separation is looking promising. The process requires simple, easy to operate and compact equipment having flexible characteristics of separation and their smooth regulation. It’s important that the process is ecologically safe [4–6]. Metal is an attractive membrane material due to its ability to dissociate molecular hydrogen at its surface. Separation through metal membrane occurs by a solution– diffusion mechanism (see Fig. 1). Hydrogen molecules from feed gas sorb on the membrane surface, where they dissociate into hydrogen atoms, then diffuse through the lattice. Hydrogen atoms emerging at permeate side of the membrane recombine K. Alhussan Space Research Institute, King Abdulaziz City for Science and Technology, Riyadh, Saudi Arabia K. Delendik (B) · N. Kolyago · O. Penyazkov · O. Voitik A. V. Luikov Heat and Mass Transfer Institute of the National Academy of Sciences of Belarus, Minsk, Republic of Belarus e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 D. Zeidan et al. (eds.), Computational Mathematics and Applications, Forum for Interdisciplinary Mathematics, https://doi.org/10.1007/978-981-15-8498-5_8

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Fig. 1 Mechanism of permeation of hydrogen through metal membranes [7]: 1—sorption, 2—dissociation, 3—diffusion, 4—reassociation, 5—desorption

with other hydrogen atoms to form hydrogen molecules and then desorb completing the permeation process. Each of these steps is characterised by an intrinsic forward and reverse step rate. The overall rate of permeation may be limited by any step which is much slower than the others or maybe governed by a combination of steps. Hydrogen flux is inversely related to the membrane thickness. Unfortunately, many metals suitable for membrane application are susceptible to both thermal and chemical degradation. The interest in palladium as membrane material stems from its relatively high permeability in comparison to other pure metals [8]. There are still significant challenges to be overcome before they can be used widely [9]. The cost of palladium increased from $400 to $2000 per troy ounce during the last decade, making it economically impractical as a membrane material for large-scale gas separation setup. Currently, membrane research is focusing on the development of new and more efficient technologies to circumvent the high capital costs of palladium. The key properties determining membrane performance are high selectivity and permeability, good mechanical, chemical and thermal stability under operating conditions, good compatibility with the operating environment, low cost and defect-free production [10]. Nickel satisfies most of these demands. It’s a face-centred cubic metal with short lattice parameters like palladium [11]. The rate-limiting step for nickel is the diffusion process. The influence of surface contamination is negligible like palladium. Therefore, this paper focuses on the mass transfer through Ni-based tubular membranes as relatively cheap metal membranes for hydrogen separation. Choice of the optimal scheme of membrane module assembling is an important task. The performance and economics of membrane process configurations should be examined and optimised via computer simulations. The aim of this study was to develop simulation algorithms and their realisation in software package for calculating mass transfer processes at hydrogen mixture membrane separation.

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2 Design of Experimental Facility for Membrane Gas Separation on the Base of Nickel Membranes Tubular configuration is the most popular because of its large packing area and low material consumption. Two cylindrical types of nickel membranes were developed at A. V. Luikov Heat and Mass Transfer Institute of the National Academy of Sciences of Belarus (Table 1). Membranes were obtained by galvanoplastics including forming matrixes copying by electrolytic deposition of nickel on them and following membranes liberation by etching of matrixes. Nickel tubular membrane produced by this technology is shown in Fig. 2. In order to increase hydrogen flow through membrane, it was decided to create binary membranes with thinned selective and compact layer and thick porous supporting one. Shown in Fig. 3, binary nickel membrane has thinned selective layer 15–20 μm thick and porous support 70–80 μm thick. The hydrogen flow through binary nickel membrane increased on 21% in comparison with hydrogen flow through homogeneous nickel membrane having the same thickness. Hydrogen flow was measured by an integral penetrability method. It was found that the activation energy of obtained nickel membranes was smaller than obtained palladium–nickel membranes, whereby the rate of hydrogen permeation was above. This fact can be explained by nanopores on membrane surface that are the centres of hydrogen sorption. Production of nickel membranes is more economical than the production of palladium-nickel membrane. Research of permeability of the individual gases (He, CH4 , N2 ) in the range of technological parameters showed that developed nickel membranes are impermeable for gases except for hydrogen. Developed nickel membranes were applied in the experimental facility for membrane gas separation (Fig. 4) that have been developed at Physical–Chemical Hydrodynamics Lab of A. V. Luikov Heat and Mass Transfer Institute of the National Academy of Sciences of Belarus. The facility was designed for studies of sepa-

Table 1 Nickel tubular and capillary membranes Characteristic Tubular membrane   3 Hydrogen permeability 3, 494 · 10−10 exp −33.1·10 RT coefficient , m3 · m/(m2 · s · Pa1/2 ) Thickness δ, μm 80 Diameter d, mm 10 Active length L, mm 270

Fig. 2 Nickel tubular membrane

Capillary membrane   3 16.73 · 10−10 exp −44.7·10 RT

150 4 270

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Fig. 3 Cross section of nickel tubular membrane

ration processes and for concentration impurity gas from hydrogen-containing gas mixtures. Developed facility for membrane separation includes six diffusion modules (Figs. 4c, 5 ) in one heat-insulating internal case; gas system: gas line of initial gases mixture, permeate gas line and retentate gas line; three internal vacuum pumps, one external vacuum pump; electrical panel with power switch, exhaust fans switch, vacuum pumps switches, vacuum relays switches, heaters switches, PID temperature controllers; ventilation: exhaust fans and louvres; external case; gas cylinders. Diffusion module (Fig. 5) can be equipped with two types of nickel membranes: tubular and capillary. Separation facility can be equipped with two types of membrane modules: membranes in own case and membranes in one case. First type—4 capillary membranes or one tubular membrane in own case (number of case in membrane module n = 6, case diameter is 14 mm); second type—membranes in one case—

Fig. 4 Experimental facility for membrane gas separation (a, b—front view, c—diffusion modules

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Fig. 5 Scheme (a) and photos (b) of diffusion module: 1—diffusion element; 2—heater; 3— collector of initial gas mixture; 4—inlet tube; 5—collector of retentate; 6—collector of permeate; 7—retentate outlet; 8—permeate outlet; 9—thermocouple; 10—reflecting barrier; 11—inside insulation; 12—external case

collector (number of tubular membranes N = 6 or number of capillary membranes N = 24). Principle of diffusion module operation is accumulative: gas mixture is fed through inlet branched tube to inlet header and apportions between diffusion elements. Gas mixture is heated up to operating temperature. In membrane module, hydrogen is concentrated in drain channel (aria with low pressure) and enriched mixture—in the delivery channel (area with high pressure). Separation facility permits operation in different regimes: • storaging regime (Fig 6a), when set separates and concentrates heavy component. Hydrogen purification takes place too. In this regime, diffusion modules are connected in parallel. • running regime (cascade) (Fig. 6b), when persistent separation of heavy gas component occurs, also hydrogen purification takes place. In this regime, diffusion modules are connected in series.

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Fig. 6 Scheme of storaging (a) and running (b) regimes of separation facility

3 Simulation of Facility for Membrane Separation Since the configuration of facility for membrane gas separation can be diverse, it’s necessary to simulate mass transfer processes. To strike a compromise between the apparatus dimensions and capacity, it is necessary to derive expressions that interrelate the geometrical and operating parameters of the apparatus. To simplify calculation next assumptions were accepted: the process of gas transfer through membrane is steady state; permeability coefficient depends on membrane characteristics, gas nature and temperature but not on pressure, does not change in the direction normal to membrane surface; separation process is isothermal.

3.1 Storaging Regime In chemical engineering the concept of the separation efficiency of an apparatus in a storaging regime is widely used as the limiting mode of the separation process [12, 13]. In this case, a module operates in the limiting mode of membrane gas separation without removing the gas mixture from the high-pressure cavity. Retentate flow rate tends to zero. In storaging system (Fig. 6a), the impurity gas is concentrated within two stages. In the first stage, the initial mixture with impurity concentration x0 and flow rate J0 is pumped in a delivery channel with volume V . Pressure in delivery channel p f is kept constant. Hydrogen passes through membrane with membrane area F into drain channel with pressure pd . After concentration reaches level x1 over time t1 , the second stage must be started: initial mixture supply is stopped, but hydrogen transfer to delivery channel goes till pressure in delivery channel decreases to selected level p2 . The concentration of impurity in the delivery channel increases little by little because of hydrogen passes through membrane. Taking into consideration that full

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gas mixing exists in the delivery channel a change of partial pressure can be described for hydrogen and impurity as follows:     RT d(1 − x) F  = J0 (1 − x0 ) − (1) p f (1 − x) − pd , pf dt V δ pf

dx RT = J0 x0 , dt V

(2)

at initial condition x = x0 at t = 0.

(3)

Impurities in the drain channel aren’t available since nickel membrane is permeable only for hydrogen at these operating parameters, i.e. selectivity α → ∞. Due to the low pressure into drain channel, there has been quite an intensive diffusion mixing in the drain channel. In the first stage, dependence of impurity concentrating time t1 on concentration can be described by formula:

⎞  ⎛ √ pd  

⎟  1−x − ⎜ ⎜√ p f ⎟ pd  √  ⎜ 2 ⎜ 1 − x − 1 − x0 + ln 

⎟ ⎟ p f  ⎝ p d ⎠   √1 − x 0 −  pf . (4) t1 (x) = − F RT  x0 δ V pf Further increasing of impurity can be achieved by simultaneous hydrogen pumping and supply flow rate stopping (J0 = 0) on the second stage. Equations 1 and 2 are altered in the forms:   RT F  d ( p (1 − x)) =− p(1 − x) − pd , dt V δ

(5)

d( px) = 0, dt

(6)

x(0) = xr 1 , p(0) = p f .

(7)

at initial conditions

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Solving equation of impurity transfer through metal membrane, Eq. 6 with initial conditions Eq. 7 relation between pressure p in delivery channel and hydrogen part in concentrated mixture is px = p f xr 1 .

(8)

In the second stage, dependence of impurity concentrating time t2 on concentration can be described by formula: √

p(1 − x) −



t2 (x) = −2

p f (1 − xr 1 ) +

√

  √ √  p(1 − x) − pd   pd ln   √   p f (1 − xr 1 ) − pd 

F RT δ V

. (9)

With a glance of Eq. 8, the expression Eq. 9 takes next form

t2 (x) = −2



   x (1 − x) pd  r 1



  −   x p xr 1 (1 − x) √ pd  f − 1 − xr 1 + ln 

 x pf  √ pd   1 − xr 1 −    pf  F RT  δ V pf

.

(10)

Total time of two stages is t = t1 + t 2 .

(11)

In the first stage, the concentrating undergoes at the expense of impurity content increasing without change of hydrogen content, in the second one—decreasing of partial pressure.

3.2 Running Regime Membrane module at running regime (Fig. 6b) works in the following way: the separated mixture is supplied into membrane module through inlet; hydrogen penetrates through membranes, goes out of the module and accumulates in the collector. Retentate is enriched by impurity and goes out of the module to the next separation stage through its collector. Relation between hydrogen flow rate and pressure difference

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is described by Sieverts–Richardson equation that is corollary of first Fick law and Sieverts equation [10, 14–16]: dJ =

    p f (1 − x) − pd dF. δ

(12)

Balance equations in the delivery channel can be described as follows: for hydrogen J0 (1 − x0 ) = J (1 − x) + J p ,

(13)

J0 x0 = J x,

(14)

J0 = J + J p .

(15)

for impurity

and total mixture flow

Gas separation is the conjugate problem of hydrogen mass transfer through metal membrane and gas mass exchange in delivery channel. Mathematical model of baromembrane separation of binary gas mixtures with account for convective external and internal membrane diffusion resistances, degree of hydrogen dissociation at adsorption on the surface of metal membrane, physical properties of gas, physicochemical properties of membrane, technological parameters of the separation process, geometry and dimensions of the membrane element was developed [10, 17]. Continuity equation, equation of motion, equation of convective diffusion are interconnected since the unknown rate of mass transfer through the membrane depends on the concentration and pressure on the delivery channel wall. The calculation data for the proposed mode l [10, 17] were located between data for the limiting cases: model of ideal mixing and model of ideal displacement. Concentration profile is equal normally to membrane surface but it can be inconstant in a parallel direction. This is true for rather high values of longitudinal velocity u and membrane length L when convective transport prevails over molecular diffusion. If Peclet number is significantly greater than 1, the regime of ideal displacement exists. Ideal mixing takes place for Pe  1 [10, 15, 18]. The experiments showed that regime of ideal mixing takes place for membranes in one case–collector, the regime of ideal displacement takes place for membranes in own case [10]. Model of ideal mixing (for membranes in one case–collector) is applicable if in delivery and drain channels concentrations and pressures are practically constant. These conditions are valid for a low stage cut (ratio of permeate flow rate and initial mixture rates) for channels with little length and diameter. In the absence of hydraulic resistance, the parameters of membrane separation are related to

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F=





J δ (x − x0 ) J0 δ (x − x0 ) √  =  √  p f (1 − x) − pd x  p f (1 − x) − pd x0

(16)

Model of ideal displacement is correct for high-speed flows and high ratio of delivery channels length and diameter when membrane stage cut and membrane selectivity are big. Gas passing through membrane is extracted by vacuum pump and pressure difference pd / p f is minor. Mass transfer is mainly carried out by convection. Membrane area is calculated by F=

J0 δx0  [ f (x0 ) − f (x)] ,  pf

(17)

where   √    1 1 + pd / p f 1 − x + pd / p f  ln 1 − 1 − x  f (x) = −  1 − pd / p f x 2 1 − pd / p f         2 pd / p f 1 − pd / p f  ln 1 + 1 − x +  ln 1 − x − pd / p f . +  1 − pd / p f 2 1 + pd / p f

3.3 Software Package “Membrane Gas Separation 1.0” (MGS V. 1.00) Simulation algorithms of membrane mass transfer processes at storaging and running regimes are implemented in software package “Membrane Gas Separation 1.0” (Fig. 7). The program operates under the control of the Windows XP/7/8/8.1/10 operating system. Editor MS Excel is required to save the results of simulations. No installation of libraries and packages is needed. Software have a user-friendly interface, advanced help system that makes it suitable for gas separation facility operators without special education. MGS v.1.0 requires the least computational effort and time in comparison with other tools such as ANSYS, OpenFOAM. The software package is designed specifically for work and research of gas separation processes on the experimental facility (Fig. 4) developed at A. V. Luikov Heat and Mass Transfer Institute of the National Academy of Sciences of Belarus. The software package operation procedure includes two stages: designing and complex calculation of mass transfer parameters (membrane area, drainage channel volume, impurity concentration in retentate, permeate volume) and saving calculation results in MS Excel editor. The first stage includes: • sinput of process operating parameters (temperature, pressure in delivery channel, pressure in drain channel);

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Fig. 7 Main Program Window

• input of impurity gas data (impurity gas, impurity concentration in the initial gas mixture); • selection of the system configuration (tubular or capillary membranes, type of diffusion module, regime, number of modules). In the area Basic data the kind and values of the input parameters are determined. Hint with allowable values is shown when moving the mouse cursor over the input cell. Operating temperature T is varied from 300◦ to 650◦ . 300 ◦ C—minimal temperature for which hydrogen penetration through membrane will be stable. 650 ◦ C— maximum temperature for which equipment will be trouble-free. Operating pressure in delivery channel p f is varied from 2.5 atm to 7 atm. 2.5 atm—minimal necessary pressure for separation; 7 atm—maximum pressure at which membrane elements will be trouble-free. Operating pressure in drain channel pd measured in atm is varied from 0.001 atm to p f . 0.001 atm—minimal pressure of vacuum pump. p f —maximum pressure at which the driving force of separation is greater than zero. Pressure difference provides the driving force of separation process, ratio of pressures in delivery and drain channels limits impurity gas concentration in the product: x 1−

pd . pf

(18)

In order to obtain maximum available concentration, it is necessary to minimise pressure in drain channels. To achieve this goal we used vacuum system—inner

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Fig. 8 Field System

tubular membrane volume was degassed. On membrane outside gas, mixture was pumped. The user chooses the desired substance of impurity gas from a list of substances (inert gases, alkanes (up to C4 ), nitrogen N2 ) in the menu located in the field Impurity gas and enters the value of impurity gas concentration at inlet to delivery channel of membrane module. Initial concentrationx0 measured in % is varied from 0.001 to 30%. In the area Membrane module the user sets the type of membrane element, type of membrane module, number of membrane modules in equipment, type of system (Fig. 8). If the user chooses Storaging system (Fig. 8a), a further field will show up on the screen, where the user can determine the number of separation stages and their duration. To change the unit of stage time, the corresponding unit has to be marked in the drop-down menu located near the area t1 or t2 . When setting the concentrationtime at the first stage t1 , from Eq. 4 is the impurity concentration in the retentate at the end of the first stage xr 1 . When setting the concentration-time at the first t1 and second t2 stages, from Eq. 10 the impurity concentration in the retentate is found at the end of the second stage xr 2 . If the user chooses the Running system (Fig. 8b), a further field will show up on the screen, where the user enters the value of retentate flow. To change the unit of retentate flow, the corresponding unit has to be marked in the drop-down menu located in the area Retentate flow. From Eqs. 16 and 17, the impurity concentrations in the retentate at the outlet are found. If all fields in the basic data area are filled with corresponding values the calculation can be started by pressing the Simulation button. The calculated result will show up in new column of the table and new curve of the plot. Table of results contains basic data (impurity gas, type of membrane element, type of membrane module, type of system, temperature, pressure in delivery channel, pressure in drain channel, initial concentration of impurity gas, number of membrane modules, for storaging system: time of the first stage for storaging system, time of the second stage for storaging system; for running system: retentate flow) and calculated results (membrane area, volume of delivery channel, for storaging system: retentate concentration at the end of the first stage, volume of penetrating hydrogen at the end of the first stage, retentate concentration at the end of the second stage, volume of penetrating hydrogen at the end of the second stage; for running system: retentate concentration). Dependence of concentrating time on impurity concentration in membrane module is shown on

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plot. The legend lists simulations plotted in the graph. Number simulation is shown. It is possible to mark one of the plotted curves of simulation by clicking on the corresponding value in the legend. It is possible to export the current simulations as an Excel file by pressing the Report button.

3.4 Verification To verify the calculation algorithm of the membrane mass transfer processes in the storaging system, the experimental data were used for the separation of nitrogen– hydrogen mixture (1% N2 /99% H2 ) into the second type of membrane module with tubular membranes nickel at T = 525 ◦ C. In the first stage of the concentration process, initial nitrogen–hydrogen mixture was pumped in the delivery channel from gas tank. The pressure in delivery channel was maintained via a reducer to 6 atm. Probes of gas mixtures were tested by chromatograph Agilent 7820A. Further, hydrogen passed through the membrane into the drain channel with pressure pd = 1 atm. After some time, the flow of the nitrogen–hydrogen mixture at inlet of the membrane module was overlapped. Simultaneously, probes of gas mixtures of the delivery channel were tested by chromatograph Agilent 7820A (Table 2). Thereafter, the membrane element was evacuated and the process was repeated. This calculated data has good correlation with the experimental one. The error lies within error chromatograph (10%). Verification confirmed the legitimacy of neglecting external diffusion resistance and using the limiting cases (model of ideal mixing and model of ideal displacement) for simulation of the process in a developed facility. The proposed calculation algorithm allows simulating of gas separation processes in the storaging and running regimes. The choice of optimum membranes, the construction of the membrane set is economically justified in the analysis of the impurity concentration. Results of simulation show that using of the regime of ideal displacement is reasonable.

Table 2 Comparison of theoretical and experimental data for impurity concentration Concentrating time (h) Experimental nitrogen Theoretical nitrogen Error (%) concentration (%) concentration (%) At storaging regime 2 3.9 2 4.4 4 7.9 4 7.5 At running regime pd = 1 bar 6 28.7 At running regime pd = 0.005 bar 6 49.4

4.3 4.3 7.6 7.6

9.3 2.3 3.9 1.3

29.1

1.4

47.0

5.1

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Table 3 Estimation of storaging system Type of mem- Tubular brane element

Capillary

Type of mem- 1 brane module

2

1

2

1

2

1

2

pd , atm

1

1

0.05

0.05

1

1

0.05

0.05

F, m2

0.31

0.31

0.31

0.31

0.49

0.49

0.49

0.49

V, l

1.27

4.49

1.27

4.49

1.27

4.76

1.27

4.76

xr 1 (6 h), %

35.40

12.12

53.54

18.12

28.08

9.01

42.53

13.33

Permeate vol- 87.08 ume, l

99.47

133.00

153.14

68.55

75.87

105.13

116.79

xr 2 , %

36.36

97.47

54.37

61.76

27.03

97.08

39.99

V /(F · 0.69 t), (l/(m2 · h)

66.99

2.45

0.69

2.45

0.43

1.62

0.43

1.62

V H2 /(F · 47.52 t), (l/(m2 · h)

54.28

72.58

83.58

23.38

25.88

35.86

39.84

4 Results and Discussion The concept of the storaging regime gives the possibility to compare the separation efficiencies of different apparatuses and to evaluate the effect of operating conditions. The separation efficiency in the storaging regime is independent of retentate rate and it depends on the characteristic of the apparatus design and the process parameters. Table 3 shows the calculating results for storaging system at T = 550 ◦ C, p f = 6 atm, x0 = 1%, t1 = 6 h. Diffusion module of the second type is more productive than module of the first type (retentane productivity for module of the first type is 0.7 l/(m2 · h) and permeate productivity for module of the first type is 48 l/(m2 · h) at pd = 1 atm, retentane productivity for module of the second type is 2.5 l/(m2 · h) and permeate productivity for module of the second type is 54 l/(m2 · h) at pd = 1 atm), but it loses in the degree of impurity concentration (nitrogen concentration is 35% for module of the first type at pd = 1 atm, nitrogen concentration is 12% for module of second type at pd = 1 atm). As shown in Table 3 and Fig. 9, retentate productivity of the module from tubular membranes is 40–60% higher as compared to the module from capillary membranes. Impurity concentration for the module from tubular membranes is 25–35% higher as compared to the module from capillary membranes. But the nickel capillary membranes have proved their application life at high temperatures and pressures (up to 10 atm and 700 ◦ C). The membrane choice allows tuning facility to get the required composition product and required efficiency.

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Fig. 9 Dependences nitrogen concentration on time for T = 550 ◦ C, p f = 6 atm, t1 = 8 h: 1—first type with tubular membrane; 2—first type with capillary; 3—second type with tubular membrane; 4—second type with capillary

Membrane module of first type with the capillary membrane is more prospective for further application (Fig. 9), especially at vacuuming of the drain channel. However, the pressure in the drain channel can’t be made as small as desired. This is due to the presence of permeate into the drain channel, to the resistance of pipelines during pumping, to the ultimate performance of the vacuum pump. Thus, 0.001 atm is the minimum value of pressure in the drain channel (minimal pressure of vacuum pump). Pressure difference provides the driving force of separation process, ratio of pressures in delivery and drain channels limits impurity gas concentration in the product. The influence of pressure in the delivery channel of membrane element on the impurity concentration varies: on the one hand, with increasing pressure in the delivery channel feed flow rate increases thereby reducing the impurity concentration; on the other hand, there is the opposite effect since increasing pressure increases the driving force for hydrogen transport through the membrane (Fig. 10). It has been established that the impurity concentration rise shifts the optimum pressure to higher values. Increase of hydrogen concentration in initial mixture results in significant growth of time in the first separation stage (Fig. 11). The optimum pressure for a minimum concentrating time in the first stage takes the form  2 p f = pd (1 − x0 )−0.5 + (1 − x)−0.5 .

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Fig. 10 Dependences of nitrogen concentration on pressure at T = 550 ◦ C, pd = 1 atm, t1 = 4 h for first type with capillary: 1—x0 = 0.5%; 2—x0 = 1%; 3—x0 = 2%

Fig. 11 Change of time of impurity concentrating in the first stage depending on heavy component concentration in initial mixture for membrane module of first a and second b types: 1—x0 = 1%, 2—x0 = 0.1%, 3—x0 = 0.05%, 4—x0 = 0.014%

Membrane mass transfer process is enhanced with increasing temperature of the gas mixture and membrane. Thus, hydrogen productivity of the membrane element is increased, membrane stage cut is increased (Fig. 12). The position of maximum impurity concentration shifts to higher values at a constant optimal pressure (Fig. 13).

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Fig. 12 Dependences of nitrogen concentration on temperature: 1—first type with tubular membrane; 2—first type with capillary; 3—second type with tubular membrane; 4—second type with capillary

Fig. 13 Dependences of nitrogen concentration on pressure in delivery channel at t1 = 4 h, pd = 1 atm, x0 = 1%: 1—T = 500 ◦ C; 2—T = 550 ◦ C; 3—T = 600 ◦ C

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5 Conclusions Nickel tubular membranes for hydrogen separation were developed. They showed good operational capability in comparison with palladium membranes at the same time being low cost. Experimental facility for membrane separation of gases on the base of nickel membranes was developed. Simulation algorithms that describe the mass transfer process of membrane gas separation were implemented in software “Membrane Gas Separation 1.0” that allows with high accuracy to carry out research of different design solutions of the experimental facility for two types of membranes (tubular and capillary) and diffusion modules (membrane in own case and membranes in one case–collector), for different regimes of gas separation (storaging and running); to choose and set the parameters of the separation process. MGS v.1.0 requires the least computational effort and time that make it suitable for gas separation facility operators without special education. The concept of the storaging regime was used to characterise gas separation membrane modules. This approach allows selecting the optimal design solution for maximum efficiency of gas separation for a specific task, and thus it’s possible to unlock the full potential of experimental facility. Results of testing proved, that developed experimental facility is suitable for hydrogen purification and impurity gas concentration. List of symbols d F J n N p R T t V x α  δ Subscripts 0 1 2 d f p r

diameter of membrane (m); membrane area (m2 ); flow rate (m3 /(m2 · s)); number of case in membrane module; number of membranes; pressure (Pa); universal gas constant R=8.31 (J/(mole · K)); operating temperature (K); time (s); volume of delivery channel (m3 ); impurity concentration (vol. fric.); selectivity; hydrogen permeability coefficient (m3 · m/(m2 · Pa0.5 · s)); membrane thickness (m) initial value; value of first stage; value of second stage; drain value; feed value; permeate value; retentate value

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Acknowledgements This research was financially supported by King Abdulaziz City for Science and Technology.

References 1. Birol, F.: The future of hydrogen. OECD (2019) 2. Sherif, S.A., Goswami, D.Y., Stefanakos, E.K., Steinfeld, A.: Handbook of Hydrogen Energy. CRC Press (2014). https://doi.org/10.1201/b17226 3. Bernardo, G., Araújo, T., da Silva Lopes, T., Sousa, J., Mendes, A.: Recent advances in membrane technologies for hydrogen purification. Int. J. Hydrogen Energy 45, 7313–7338 (2020). https://doi.org/10.1016/j.ijhydene.2019.06.162 4. Uragami, T.: Science and Technology of Separation Membranes. Wiley, Chichester, UK (2017). https://doi.org/10.1002/9781118932551 5. Ismail, F.A., Hilal, N., Wright, C.J. (eds): Memrane Fabrication. CRC Press, Taylor & Francis Group (2015) 6. Li, P., Wang, Z., Qiao, Z., Liu, Y., Cao, X., Li, W., Wang, J., Wang, S.: Recent developments in membranes for efficient hydrogen purification. J. Membr. Sci. 495, 130–168 (2015). https:// doi.org/10.1016/j.memsci.2015.08.010 7. Baker, R.W.: Membrane Technology and Applications. Wiley (2004) 8. Doukelis, A., Panopoulos, K., Koumanakos, A., Kakaras, E. (eds.): Palladium Membrane Technology for Hydrogen Production, Carbon Capture and Other Applications. Elsevier (2015) 9. Al-Mufachi, N.A., Rees, N.V., Steinberger-Wilkens, R.: Hydrogen selective membranes: a review of palladium-based dense metal membranes. Renew. Sustain. Energy Rev. 47, 540–551 (2015). https://doi.org/10.1016/j.rser.2015.03.026 10. Alhussan, K., Delendik, K., Ignatenko, D., Kolyago, N., Penyazkov, O., Voitik, O.: Basic Principles of Membrane Gas Separation. HMTI NANB, Minsk (2015) 11. Gapontsev, A.V., Kondrat’ev, V.V.: Hydrogen diffusion in disordered metals and alloys. Phys. Uspekhi. 46, 1077–1098 (2003). https://doi.org/10.3367/UFNr.0173.200310c.1107 12. Vorotyntsev, V.M., Drozdov, P.N., Vorotyntsev, I.V., Belyaev, E.S.: Deep gas cleaning of highly permeating impurities using a membrane module with a feed tank. Pet. Chem. 51, 595–600 (2011). https://doi.org/10.1134/S0965544111080111 13. Vorotyntsev, V.M., Drozdov, P.N., Vorotyntsev, I.V., Balabanov, S.S.: Membrane module with a feed tank for fine purification of gases. Theor. Found. Chem. Eng. 42, 398–403 (2008). https:// doi.org/10.1134/S0040579508040076 14. Caravella, A., Scura, F., Barbieri, G., Drioli, E.: Sieverts law empirical exponent for PD-based membranes: critical analysis in pure H2 permeation. J. Phys. Chem. B. 114, 6033–6047 (2010). https://doi.org/10.1021/jp1006582 15. Luis, P.: Fundamental Modeling of Membrane Systems. Elsevier Inc. (2012) 16. Deveau, N.D., Ma, Y.H., Datta, R.: Beyond Sieverts’ law: a comprehensive microkinetic model of hydrogen permeation in dense metal membranes. J. Membr. Sci. 437, 298–311 (2013). https://doi.org/10.1016/j.memsci.2013.02.047 17. Baikov, V.I., Primak, N.V.: Membrane selective separation of binary gas mixtures. J. Eng. Phys. Thermophys. 80, 382–387 (2007). https://doi.org/10.1007/s10891-007-0050-8 18. Mulder, M.: Basic Principles of Membrane Technology (1996)

An Efficient Hybrid Conjugate Gradient Method for Unconstrained Optimisation Talat Alkhouli, Hatem S. A. Hamatta, Mustafa Mamat, and Mohd Rivaie

Abstract Optimisation refers to a common procedure applied within the science and engineering domain to determine variables that produce the best performance values. One of the common efficient techniques to solve large-scale unconstrained optimisation issues is the conjugate gradient method, because of its simplicity, low memory consumptions and global convergence properties. This method embeds the n-step to attain a minimum point, where convergence properties are absent. Several techniques do not perform well according to the number of iteration and CPU time. In order to address these shortcomings, this study proposed new Hybrid CG coefficients, βk , Tal’at and Mamat (HTM). The proposed parameter βkH T M is computed as a combination of βkH S (Hestenes–Steifel formula), βkL S (Liu–Storey formula) and βkR M I L (Rivaie formula) to exploit attractive features of each. The algorithm uses the exact line search. Numerical results and their performance profiles display that the proposed method is promising. It is also shown that’s the new formula for βk performs much better than the original Hestenes–Steifel, Liu–Storey and the Rivaie methods.

T. Alkhouli · H. S. A. Hamatta (B) Department of Applied Sciences, Aqaba University College, Al-Balqa Applied University, Aqaba, Jordan e-mail: [email protected] T. Alkhouli e-mail: [email protected] M. Mamat Faculty of Informatics and Computing, Department of Computer Science and Mathematics, University Sultan Zainal Abidin, 21030 Terengganu, Malaysia e-mail: [email protected] M. Rivaie Department of Computer Science and Mathematics, Univesiti Technology MARA (UITM), 23000 Terengganu, Malaysia © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 D. Zeidan et al. (eds.), Computational Mathematics and Applications, Forum for Interdisciplinary Mathematics, https://doi.org/10.1007/978-981-15-8498-5_9

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1 Introduction Optimisation is an important characteristic of our daily lives, as obvious in how we need to make the choice that yields maximum outcome in any goal we have in real life. This is especially essential in many specific areas like economy, management, science and engineering. For example, in engineering, the use of optimisation is necessary for determining the best design, like the design for the lowest cost building or the design to maximise the amount of load in a moving vehicle. A suitable optimisation process is also particularly important in marketing in order to maximise the business product. It might mean in business maximising the profit, minimising the loss or minimising the risk. Optimisation could be well defined as the science that determines the ideal solution to a specific mathematically defined model problem. In other words, the procedures used to make a design or a system as efficient as possible. It helps in improving the quality of decision- making. That might mean in engineering maximising the strength of a bridge or minimising the weight of that bridge and so on. Therefore, considerate and use the concept of optimisation is essential and gainful at the same time. Conjugate gradient methods (CGMs) are very effective for solving the large-scale unconstrained optimisation problem. Because of its global convergence properties and small memory requirement, the CGM became one of the best methods in unconstrained optimisation. The widespread and increasing use of optimisation makes it an important subject to study for students and specialists in the most well-known subjects. Numerous changes have been made in the past years in order to develop the method. As a result, hundreds of modification algorithms have been confirmed. Nevertheless, some of them have complicated formulation and global convergence problems. The CGMs were presented to minimise quadratic functions before it was expanded to solve the unconstrained optimisation problem for non-linear functions. In finite number of iterations, it has the ability to find the minimiser of a quadratic function. Methods that have the ability to solve a quadratic function are greatly evaluated since they are expected to perform well on other non-quadratic functions in the neighbourhood of a local minimum. Today, even though the many different methods that had been generated to tackle the increasing effort of those objective functions, there are still accommodations for more development to guarantee higher effectiveness. On the other side, there is an exceedingly various choice of real-world applications and it is practically impossible to improve a single algorithm that will solve all optimisation problems efficiently. Henceforth, there have been many modification methods to solve different kinds of problems. To this day, researchers try to find the best CGM that has all the essential criteria like efficiency, robustness, easy to use and accurate. The aim of this study is to modify a new CGM that is easy to implement and comparable with the original CG methods. In order to do that the new modification of CG has to fulfil sufficient descent condition. Moreover, efficiency and good numerical performance as regards to real computation. The ability to solve problems on large

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scale robustly is also needed. CGMs were primarily designed to solve problems in the form: min f (x), x ∈ Rn

(1)

where f : R n → R is a differentiable function. The optimisation problem in Eq. (1) can be categorised as a decision problem that gathers the ‘best’ vector x of the decision variables over all probable vectors in R n . The ‘best’ vector, means the one that provides the minimum value to the objective function. In any unconstrained optimisation problem, the objective function is minimised to its optimal point x ∗ starting with an initial point that is near to it or around its neighbourhood. In applied mathematics, a local optimum of an optimisation problem is a solution that is optimum within a neighbouring set of candidate solutions. This is indifferent with a global optimum solution, which is the optimum solution for all possible solutions, not just the ones in a particular neighbourhood of values. Optimisation methods with local property need suitable initial points as they will seriously affect the objective value of the needed solution. When the global minimiser is found, the optimisation problem is solved. The form of an iterative method to solve unconstrained optimisation problem is given by xk+1 = xk + αk dk k = 0, 1, 2

(2)

where xk is the current iterate, αk is the positive step size realised by performing a single-dimensional search, known as ‘line search’. The step size αk decides how far the function is to be minimised in each iteration. In other words, the αk is used to adopt how far an initial point can be reduced in the direction of its next iteration until the solution point is found. The aim of using different choices of αk is basically to guarantee that the formula defined by Eq. (2) is globally convergent. The most common one is the exact line search which is f (xk + αk dk ) = min f (xk + αdk ) α≥0

and dk is the search direction defined by  −gk , if k = 0; dk = −gk + βk dk−1 , if k ≥ 1

(3)

(4)

where βk is considered as a parameter and gk is the gradient of f(x) at xk . In the linear CGMs or non-linear CGMs, the parameter βk is called the conjugate gradient coefficient [1]. Different choices of βk will yield a different CGM. Table 1 arranges a sequential list of the six most commonly CGMs used in research papers worldwide and recognised as the best well known. The CD, FR and DY methods with gkT gk in the numerator of their βk have strong convergence properties, but they are vulnerable to jamming. More simply, they start

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Table 1 Various choices for the classical CG parameter Choice References g T (gk−g ) = (g k−g )Tk−1 k k−1 dk−1 gT g βkF R = T k k gk−1 gk−1 g T (gk−gk−1 ) P R P βk = k T gk−1 gk−1 g T gk βkC D = T k dk−1 gk−1 gkT (gk −gk−1 ) L S βk = T g dk−1 k−1 gT g βkDY = (g −g k )kT d k k−1 k−1

βkH S

Eq. No.

[2]

(5)

[3]

(6)

[4, 5]

(7)

[6]

(8)

[7]

(10)

[8]

(11)

taking small steps without making any substantial progress. On the other hand, LS, HS and PRP methods with gkT (gk − gk−1 ) in the numerator of their βk have built-in restart feature that prevents the jamming phenomenon. When the step αk is small, the factor gk − gk−1 in the numerator of their βk tends to zero. There is frequent research on the convergence properties of these methods. Zoutendijk [1] managed to prove the global convergence property of the FR method. In 1985, Al-Baali [9] succeeded in implementing the first FR process of global convergence under the Strong Wolfe condition. Powell [10] improved the PRP method to create only non-negative values of βk . Dai and Yuan [11] studied the global convergence of the CD method under the strong Wolfe line search. Dai and Yuan also have revealed that the DY method is convergent with Goldstein and Armijo line searches. The global convergence property of the FR method for non-quadratic objective functions was proved, when the Strong Wolfe line search was used [3]. The PRP method may not converge globally under some traditional line searches. Some convergent versions were proposed by using some new complicated line searches, or by restricting the parameter to a non-negative number [12]. The CD method and DY method were proved to converge globally under the Strong Wolfe line search [13, 14]. However, the global convergence of PRP, LS and HS methods have not been established under all types of line searches. The main reason is that many CGMs cannot guarantee that the descent property occur at each iterative. In the latest years, based on the above formulas and their hybridisation, many works are putting effort into seeking new CGMs with not only good convergence property but also excellent numerical effects were published. Nazareth [15] regarded the FR, PRP, HS and DY formula as the four leading contenders for the scalar βk and proposed two-parameter family of conjugate gradient method. βk =

λk ||gk ||2 + (1 − λk )gkT yk−1 T μk ||gk−1 ||2 + (1 − μk )dk−1 yk−1

(5)

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where λk , μk ∈ [0, 1] are parameters, yk−1 = gk − gk−1 . Wei et al. [14] introduced a modification of the FR method which is called the VFR method is given by βkV F R =

μ1 ||gk ||2 T μ2 |gk dk−1 | + μ3 ||gk−1 ||2

(6)

where μ1 ∈ (0, +∞), μ2 ∈ (μ1 + ε1 , +∞), μ3 ∈ (0, +∞) and ε1 is any given positive constant. Huang et al. [16] modified LS to βkm L S =

gkT (gk − tk gk−1 ) T T u|dk−1 gk | − dk−1 gk−1

(7)

k || where tk = ||g||gk−1 , u>0 || Another famous CGM is the RMIL method, related to the researchers: Rivaie et al. [17]. Like HS and PRP, RMIL retained the original numerator to give it restart property and designed a new formula for the denominator in order to improve its behaviour. Its CG coefficient is written as

βkR M I L =

gkT (gk − gk−1 ) T dk−1 dk−1

(8)

Dai [13] modified HS and suggested taking βk∗ =

η||gk ||2 , η ∈ (σ, 1] T gkT dk−1 − ηgk−1 dk−1

(9)

Zhang extended the result of the HS [18] method and suggested the NHS method as follows: βkN H S =

||gk || |g T g | ||gk−1 || k k−1 ||gk−1 ||2

||gk ||2 −

(10)

Within the past six decades, since CG was first introduced into the optimisation field, a lot of modifications were done by previous researchers to improve its overall performance. Presently, there are many other formulas of the CGMs; each comes in more complex form compared to the classical CG. The idea of the hybrid method has long taken a basis in the studies of unconstrained optimisation. Some classical CGMs have strong convergence properties like FR, DY and CD, but they may not perform well. Others like PRP, HS and LS may not converge but they do well. So hybrid conjugate gradient algorithms have been designed to use and combine the attractive features of the classical conjugate gradient algorithms. The algorithms of the hybrid CG method combine two or more classical CGMs for the purpose of avoiding the jamming problem and in order to apply their interesting behaviours.

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The hybrid CGMs are principally devised based on an adaptive switch from a CGM in the first category to one or more in the second category when the iterations jam. Generally, the performances of the hybrid CGMs are better than the performances of the classical CGMs. Listed below are some well-known hybrid CG methods. This reason leads Powell [10] to modify the PRP method by βkP R P+ = max{βkP R P , 0}

(11)

By the same motivations, Touati-Ahmed and Storey [19] enhance AL-Baali’s [9] convergence result on the FR method βk ∈ {0, βkF R }

(12)

to become  βkT S =

βkP R P , if 0 ≤ βkP R P ≤ βkF R ; βkF R , elsewhere

(13)

The TS method has nice convergent properties under the exact line search and has exposed improvement in its numerical performance compared to FR and PRP. Hu and Storey [20] suggest the formula βkH HU S = max{0, min{βkP R P , βkF R }}

(14)

The PRP method has a built-in restart property in which the jamming problem would be automatically detected since it will restart repeatedly if a bad direction occurred. Consequently, in this combination of PRP and FR when a negative value is obtained, the value of the βk returns to zero instead as the negative value lead to slower descent of the algorithm. Therefore, the HHUS method has some good properties of both the PRP method and the FR method and showed it to be more effective than some modified CGMs and other hybrid CGMs. Gilbert and Nocedal [21] extended the formula of (14) to the interval βk ∈ {−βkF R , βkF R } and propose the formula βkH G N = max {−βkF R , min{βkP R P , βkF R }}

(15)

The performance of the HGN method according to the numerical results reported in [21] was not better than that of PRP+. With this special adjustment, βkH G N can be negative since βkF R is always non-negative. The HGN hybrid method is the best overall performance of both the PRP method and the FR method and some other hybrid CG algorithms.

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Dai and Yuan [11] proposed a family of globally convergent conjugate method as βk =

λ||gk−1

||2

||gk ||2 T + (1 − λ)dk−1 yk−1

where λ ∈ [0, 1] is parameter, yk−1 = gk − gk−1 . In 2015, Alhawarat et al. [22] depict a new hybrid of conjugate gradient method which relates to PRP formula ⎧ T ⎨βkP R P = gk (gT k −gk−1 ) if ||gk ||2 < |gkT gk−1 | gk−1 gk−1 P R P∗∗ ||g || βk = 2 k ⎩β N P R P = ||gk || − ||gk−1 || |gk gk−1 | , elsewhere k ||gk−1 ||2 Recently, Xiao Xu and Fan-yu Kong [23] make a combination with parameters βk of the DY method and the HS method, and propose the hybrid method:  a1 βkDY + a2 βkH S if ||gk ||2 < |gkT gk−1 | 1 βk = 0 elsewhere A new hybrid CG is considered by Djordjevi´c [24]. The CG parameter βk is computed as a convex combination of βkC D and βkL S : hyb

βk

= (1 − θk ).βkL S + θk βkC D

where the conjugacy condition is satisfied if the parameter θk is computed in such a way. More recently, Yasir [25] proposed a new hybrid CG similar to PRP as follows: ⎧ T ) ⎨ gk (gk −gk−1 if ||gk ||2 < |gkT gk−1 | ||g ||2 ||gk || βkY H M = gkT (gkk−1 − ||g || gk−1 ) ⎩ k−1 , otherwise ||gk−1 ||2 which is a hybrid of PRP and WYL methods. This method satisfies the sufficient descent condition and global convergence properties under an exact line search.

2 New Hybrid CGM In recent years, a great deal of effort has been made to develop new modifications of CGMs, as we mentioned earlier, which not only have strong convergence properties but are also computationally superior to the well-known methods. As a result of that, hundreds of modifications, CG algorithms have been confirmed. Andrei gives a survey of 40 non-linear CG algorithms for unconstrained optimisation [26].

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The hybrid method is one of the widespread methods for modifying the CG method. The idea is to combine two or more different CG algorithms to form a new hybrid algorithm in order to use their attractive features and apply their interesting behaviours. In this section, enlightened by above-mentioned ideas [2, 21], we suggest our βk which named as βkH T M . Where HTM represents Hybrid Tala’t and Mustafa. βkH T M = max{βkR M I L , min{βkL S , βkH S }}

(16)

The procedure begins with a starting value that is substituted into the formula. The outcome is then taken as the new starting point which is then substituted into the formula again. And so on the process continues to repeat until the solution point is reached. The algorithm is given as follows: Algorithm 2.1 Step 1: Initialisation. Given x0 ∈ R n , ε ≥ 0 set d0 = −g0 i f ||g0 || ≤ ε then stop. Step 2: Compute αk by Eq. (3). Step 3: Let xk+1 = xk + αk dk , gk+1 = g(xk+1 ) i f ||gk+1 || ≤ ε then stop. Step 4: Compute βk by (16), and generate dk+1 by Eq. (4). Step 5: set k = k + 1 go to Step 2.

3 Global Convergence Properties The convergent properties of βkH T M will be studied in this section. We prove the result of convergence for the common CGM only. To verify the convergence, we assumed that every search direction dk fulfil the descent condition gkT dk < 0

(17)

for all k > 0. If there exists a constant λ > 0 for all k > 0 then, the search directions satisfy the following sufficient descent condition: gkT dk < −λ||gk ||2

(18)

The following Theorem is very important in the establishment of a sufficient descent condition. Theorem 1 Consider a CGM with the search direction (4) and βkH T M given as (16) then condition (18) holds for all k > 0. Proof If k = 0 then it is clear that g0T d0 = −λ||gk ||2 . Hence, condition (18) holds true. We also need to show that for k ≥ 1, condition (18) will also hold true.

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T From (4), multiply both sides by gk+1 , we obtain T T dk+1 = gk+1 (−gk+1 + βk+1 dk ) gk+1

T = −||gk+1 ||2 + βk+1 gk+1 dk T For exact line search, we know that gk+1 dk = 0. Thus, T dk+1 = −||gk+1 ||2 gk+1

Therefore, it implies that dk+1 is a sufficient descent direction. Hence, gkT dk ≤ −λ||gk ||2 holds true. The proof is completed.



4 Numerical Result and Discussion In order to check the efficiency of HTM, the researcher compares HTM method with all well-known methods. Table 2 shows the computational performance of MATLAB R2015a program on a set of unconstrained optimisation test problems. This software is known to offer high performance in numerical computations, as well as its coding language and interface that are both easy to understand and to apply. We select randomly 25 test functions from Andrei [27]. A part of the numerical computations, a code programming is developed by using MATLAB R2015a software by implementing a 25 set of unconstrained optimisation test problems involve convex and non-convex functions to see the behaviour of the algorithm in different situations such as single global minimum and multiple or single global minima in the presence of many local minima. Difficult situations like functions with significant null-space effects, long narrow valleys, essentially unimodal functions and functions with a huge number of significant local optima. In order to evaluate the efficiency and robustness of the modify CGM, the numerical performance of the proposed CGM is compared with the well-known CGMs. The results are taken in terms of the number of iteration (NOI) and Computational Processing Unit (CPU) times. Then, the performance profile technique is used to evaluate the performance of all the experimental methods. The graphs of performance profiles are created by Sigma Plot software.

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Table 2 List of problem functions No Function Dim

Initial point

1

SIX HUMP CAMEL

2

(−1, −1), (3, 3), (50, 50)

2

TRECCANI

2

(0.5, 0.5), (15, 15), (150, 150)

3

ZETTL

2

(−2, −2), (0.3, 0.3), (5, 5)

4

QUARTIC

4

(10, …, 10), (50, …, 50), (100, …, 100)

5

EXTENDED HIMMELBLAU

4

(−4, …, −4), (−1.5, …, −1.5), (1, …, 1)

6

EXTENDED MARTOS

10

(−2, …, −2), (0.5, …, 0.5), (2, …, 2)

7

QUADRATIC QF2

100, 500, 1000

(1, …, 1), (15, …, 15), (60, …, 60)

8

GENERALZED QUARTIC

100, 500, 1000

(−0.5, …, −0.5), (1, …, 1), (6, …, 6)

9

WHITE AND HOLST 100, 500, 1000

(−2, …, −2), (2, …, 2), (9, …, 9)

10

FLETCHCR

100, 500, 1000

(−4, …, −4), (3, …, 3), (11, …, 11)

11

ROSENBROCK

100, 500, 1000

(5, …, 5), (25, …, 25), (30, …, 30)

12

EXTENDED DENSCHNB

100, 500, 1000

(1, …, 1), (16, …, 16), (25, …, 25)

13

EXTENDED BEALE 100, 500, 1000

(0.5, …, 0.5), (2, …, 2), (11, …, 11)

14

EXTENDED TRIDIAGONAL

(3, …, 3), (9, …, 9), (50, …, 50)

15

DIAGONAL4

100, 500, 1000

(0.2, …, 0.2), (60, …, 60), (200, …, 200)

16

SUM SQUARES

100, 500, 1000

(−1, …, −1), (60, …, 60), (150, …, 150)

17

SHALOW

100, 500, 1000

(0.2, …, 0.2), (3, …, 3), (30, …, 30)

18

PERTURBD QUADRATIC

100, 500, 1000

(0.5, …, 0.5), (2, …, 2), (12, …, 12)

19

DIXON AND PRICE

100, 500, 1000

(0.2, …, 0.2), (0.4, …, 0.4), (16, …, 16)

20

QUADRATIC QF1

100, 500, 1000

(1.5, …, 1.5), (5, …, 5), (20, …, 20)

21

NONDIA

100, 500, 1000

(3, …, 3), (7.5, …, 7.5), (50, …, 50)

22

DQDRTIC

100, 500, 1000

(10, …, 10), (60, …, 60), (100, …, 100)

23

SINQUAD

100, 500, 1000

(4, …, 4), (20, …, 20), (60, …, 60)

24

GENERALIZED QUARTIC GQ2

100, 500, 1000

(0.5, …, 0.5), (15, …, 15), (25, …, 25)

25

EXTENDED QUADRATIC PENALTY QP2

100, 500, 1000

(1, …, 1), (10, …, 10), (50, …, 50)

100, 500, 1000

In this study, we choose ε = 10−6 and stopping criteria is defined as ||gk || ≤ ε as Hillstrom [28] recommended. From a point closer to the solution point to a point far away from the solution point, three initial points are chosen so that they can be used to test the global convergence of the new CG coefficient. The dimensions n of 25 problems are 2, 4, 10, 100, 500 and 1000.

An Efficient Hybrid Conjugate Gradient Method for Unconstrained Optimisation

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In some cases, the calculations were blocked due to the failure of the line search to find the positive step size and were, therefore, considered to have failed. Numerical outcomes are compared to the number of iterations and the time of the CPU. We use the performance profile presented by Dolan and More [29] to get the performance results shown in Figs. 1, 2 and 3. The CPU processor used was Intel (R) Core TM i3-M350 (2.27GHz), with RAM 4 GB. The subroutine computer codes pertain to the exact line search formed by the following MATLAB codes: Program Code function [x,k,t,f]=CG(x0,OP1,OP2) clear t; tic \%This function using (CG) methods for finding \%the minimizer of a given function by using the Exact line search \%Outputs: \% x: The point that the minimum of the test function occurs at it. \% k: The number of iterations. \% t: CPU time.clc \%Inputs:\\ \% x0: initial point. \% OP1: The CG method.clear all \% OP2: A number for a test function. g=test_functions(x0,OP2,1); f=test_functions(x0,OP2); d=-g; k=0; x=x0; %format long e while (norm(g)>.000001) && (k1000) in the analysis. iv. Use other approaches for modifications such as three-term CG method. These are the recommendations for upcoming research. Acknowledgements The author would like to thank ZARQA UNIVERSITY and the (IACMC2019) ORGANIZING COMMITTEE for funding this study. We are also grateful to UNISZA for their considerations and comments.

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References 1. Zoutendijk, G.: Nonlinear programming, computational methods. Integer and Nonlinear Programming, pp. 37–86 (1970) 2. Magnus Rudolph Hestenes and Eduard Stiefel: Methods of Conjugate Gradients for Solving Linear Systems, vol. 49. NBS Washington, DC (1952) 3. Fletcher, R., Reeves, C.M.: Function minimization by conjugate gradients. Comput. J. 7(2), 149–154 (1964) 4. Boris Teodorovich Polyak: The conjugate gradient method in extremal problems. USSR Comput. Math. Math. Phys. 9(4), 94–112 (1969) 5. Polak, E., Ribiere, G.: Note sur la convergence de directions conjugées. rev. francaise informat. Recherche Opertionelle, 3e année, 16, 35–43 (1969) 6. Fletcher, R.: Practical Methods of Optimization. Wiley (2013) 7. Liu, Y., Storey, C.: Efficient generalized conjugate gradient algorithms, part 1: theory. J. Optim. Theory Appl. 69(1), 129–137 (1991) 8. Dai, Y.-H., Yuan, Y.: A nonlinear conjugate gradient method with a strong global convergence property. SIAM J. Optim. 10(1), 177–182 (1999) 9. Al-Baali, M.: Descent property and global convergence of the fletcher–reeves method with inexact line search. IMA J. Num. Anal. 5(1), 121–124 (1985) 10. Powell, M.J.D.: Nonconvex minimization calculations and the conjugate gradient method. Numerical Analysis, pp. 122–141. Springer (1984) 11. Dai, Y., Yuan, Y.: A class of globally convergent conjugate gradient methods. Sci. China Ser. A Math. 46(2), 251 (2003) 12. Li, M., Feng, H.: A sufficient descent ls conjugate gradient method for unconstrained optimization problems. Appl. Math. Comput. 218(5), 1577–1586 (2011) 13. Dai, Z.-F.: Two modified hs type conjugate gradient methods for unconstrained optimization problems. Nonlinear Anal. Theory Methods Appl. 74(3), 927–936 (2011) 14. Wei, Z., Li, G., Qi, L.: New nonlinear conjugate gradient formulas for large-scale unconstrained optimization problems. Appl. Math. Comput. 179(2), 407–430 (2006) 15. Nazareth, J.L.: Conjugate-gradient methods. In: Floudas, C., Pardalos, P. (eds.) Encyclopedia of Optimization (1999) 16. Huang, H., Wei, Z., Shengwei, Y.: The proof of the sufficient descent condition of the WeiYao-Liu conjugate gradient method under the strong Wolfe-Powell line search. Appl. Math. Comput. 189(2), 1241–1245 (2007) 17. Rivaie, M., Mamat, M., Leong, W.J., Ismail, M.: A new conjugate gradient coefficient for large scale nonlinear unconstrained optimization. Int. J. Math. Anal. 6(23), 1131–1146 (2012) 18. Zhang, L.: New versions of the Hestenes-Stiefel nonlinear conjugate gradient method based on the secant condition for optimization. Comput. Appl. Math. 28(1) (2009) 19. Touati-Ahmed, D., Storey, C.: Efficient hybrid conjugate gradient techniques. J. Optim. Theory Appl. 64(2), 379–397 (1990) 20. Hu, Y.F., Storey, C.: Global convergence result for conjugate gradient methods. J. Optim. Theory Appl. 71(2), 399–405 (1991) 21. Jean Charles Gilbert and Jorge Nocedal: Global convergence properties of conjugate gradient methods for optimization. SIAM J. Optim. 2(1), 21–42 (1992) 22. Alhawarat, A., Mamat, M., Rivaie, M., Salleh, Z.: An efficient hybrid conjugate gradient method with the strong wolfe-powell line search. Math. Problems Eng. (2015) 23. Xiao, X., Kong, F.: New hybrid conjugate gradient methods with the generalized wolfe line search. SpringerPlus 5(1), 881 (2016) 24. Djordjevic, S.: New hybrid conjugate gradient method as a convex combination of ls and cd methods. Filomat 31(6) (2017) 25. Salih, Y., Hamoda, M.A., Rivaie, M.: New hybrid conjugate gradient method with global convergence properties for unconstrained optimization. Malays. J. Computi. Appl. Math. 1(1), 29–38 (2018)

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26. Andrei, N.: Open problems in nonlinear conjugate gradient algorithms for unconstrained optimization. Bull. Malays. Math. Sci. Soc. 34(2) (2011) 27. Andrei, N.: An unconstrained optimization test functions collection. Adv. Model. Optim. 10(1), 147–161 (2008) 28. Hillstrom, K.E.: Simulation test approach to the evaluation of nonlinear optimization algorithms. ACM Trans. Math. Softw. (United States) 3(4) (1977) 29. Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91(2), 201–213 (2002)

On the Polynomial Decay of the Wave Equation with Wentzell Conditions Karima Laoubi and Djamila Seba

Abstract We consider the wave equation on the unit square of the plane with feedback, satisfying the classical geometric control condition. It is well known that the resolvent operator associated with this equation must be unbounded on the imaginary axis which implies that there is no exponential stability in the energy space. The purpose of this work is to show that the solution of this system decays polynomially by using a one-dimensional wave equation subject to Wentzell-Dirichlet boundary conditions combined with Fourier analysis and spectral theory.

1 Introduction This chapter is devoted to studying the stabilisation of the linear wave equation with Wentzell dynamic boundary condition in a bounded domain. Wentzell-type boundary conditions for differential operators were introduced in 1959 by Wentzell in the context of diffusion processes for the heat equation and appear naturally in the multidimensional diffusion processes in the pioneering work of Wentzell [20, 30] (see also Feller’s work for one-dimensional processes [10, 11]). They can also be derived as approximate boundary conditions in asymptotic problems or artificial boundary conditions in exterior problems, see, for example, [8, 14, 15, 19, 27] and the references contained therein. It is well known [14, 15] that the natural feedback used in our problem is not sufficient to guarantee an exponential decay. Hence, we are interested in proving a weaker decay of the energy. More precisely we will establish sufficient conditions and methods that guarantee a polynomial decay of the energy of our system. K. Laoubi · D. Seba (B) Dynamic of Engines and Vibroacoustic Laboratory, Univerty M’Hamed Bougara of Boumerdes, Boumerdes, Algeria e-mail: [email protected] K. Laoubi e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 D. Zeidan et al. (eds.), Computational Mathematics and Applications, Forum for Interdisciplinary Mathematics, https://doi.org/10.1007/978-981-15-8498-5_10

209

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K. Laoubi and D. Seba

The proof is based on spectral theory, the multiplier technique and a specific Fourier analysis of the obtained one-dimensional problem combined with Ingham’s inequality (see [26]). Various approaches have been adopted to deal with the boundary stabilisation of the polynomial decay rate: An energy inequality was established in [1, 25, 29], a semigroup theory was used in [2, 4–6], a Fourier analysis method was presented in [21, 33], a perturbation method in [13, 32], an optimal observability inequality was investigated in [9], a logarithmic estimate of the resolvent can be found in [23]. Another method by using Ingham’s inequality was formulated in [3] and the references therein. This chapter is divided into four sections. In Sect. 2, the mathematical model is presented. Also, it is proved that the studied problem is well defined. The third Section presents the analysis of the one-dimensional wave dissipative equation. The polynomial stability of the Wentzell problem, using Fourier analysis, is presented in the last section.

2 The Problem Let  = (0, 1)2 ⊂ R2 be an open-bounded domain with smooth boundary ∂ = . We suppose that  is composed of two disjoint parts  D and V defined by V = {(1, y) : 0 < y < 1} ∪ {(0, y) : 0 < y < 1} = V1 ∪ V2 ,  D = ∂\1 = {(x, 0) : 0 < x < 1} ∪ {(x, 1) : 0 < x < 1} =  1D ∪  2D Let

⎧ z tt − z = 0 ⎪ ⎪ ⎨ ∂η z −  T z + z t = 0 z=0 ⎪ ⎪ ⎩ z (., 0) = z 0 , z t (., 0) = z 1

in on on in

 × R+ , V × R+ ,  D × R+ , ,

(1)

T means the tangential Laplace operator on V [30]. In our case, T = ∂ y y , ∂η z = ∂x z z = z(x, y, t) where (x, y) ∈ , t ∈ R+ , η is the unit outward normal vector along the boundary. Let  X = ϑ ∈ H 1 () : ϑ| D = 0, ϑ|V ∈ H01 (V )}, be a Hilbert space with norm  ϑ2X =

 

|∇ϑ|2 d x +

V

|∇T ϑ|2 d

(2)

On the Polynomial Decay of the Wave Equation with Wentzell Conditions

211

∇T denotes the tangential derivative along V . We define H = X × L 2 () the Hilbert space with the inner product H given by   < (z, ϑ) , z¯ , ϑ¯ > H = (∇z, ∇ z¯ ) L 2 () + (∇T z, ∇T z¯ ) L 2 (V ) + ϑ, ϑ¯ L 2 ()

(3)

1/2

and associated norm:  ·  H =< ·, · > H . We note V ι , the common points of V with the vertices of , ι = 1, 2, 3, 4.  For z ∈ X , we have T (z/ V ι ) ∈ H −1 (V ι ) = (H01 (V ι )) . For z ∈ X with z ∈

21 (V ι )) , hence ∂η z ∈ H −1 (V ι ) L 2 (), using a result of [12], we have ∂η z ∈ ( H 1 

2 (V ι )) ⊂ H −1 (V ι ). Let us now define a linear unbounded operator A because ( H on H by A(z, ϑ) = (ϑ, z) (4) with domain  D (A) = (z, ϑ) ∈ X × X, u ∈ L 2 () : ∂η z − T z + ϑ = 0 on V ι , ι ∈ {1, 2, 3, 4}}

Note that the above expression, ∂η z − T z + ϑ = 0 on V ι , is well defined on H −1 (V ι ). By defining the energy E(t) = E(z, z t )(t) =

1  (z, z t ) (t) 2H 2

(5)

For (z 0 , z 1 ) ∈ D (A) and integrating by part, we get that d E (t) = − dt

 V

|z t |2 d

(6)

This shows that the system (1) is dissipative. In this formalism, the system (1) reads as an evolution equation

Yt (t) = AY (t) Y (0) = Y0

(7)

with Y = (z, z t ), Y0 = (z 0 , z 1 ). We can now state the existence and uniqueness result for the associated evolution equation [7]. Theorem 1 ) ∈ D (A), then there exists a unique (z, ϑ) ∈ C 

1 If (z 0 , z 0, ∞ , D (A) ∩ C1 0, ∞ , H that satisfies the Cauchy problem (7).

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Proof The proof is based on the Lumer-Phillips’s theorem. We start with the dissipatedness. Let U = (z, ϑ) ∈ D(A). Using (3) and (4), we obtain  AU, U H =



 ∇z∇ϑd x +

 V

∇T z∇T ϑd +



z.ϑd x

Then, integrating by parts and using the boundary conditions, we get  ReAU, U H = −

|ϑ|2 d ≤ 0

V

(8)

Let us pass to the maximality. Letting f = ( f 1 , f 2 ) ∈ H, we look for a unique element U = (z, ϑ) ∈ D(A) such that −AU = f . Equivalently, we get −ϑ = f 1 and − z = f 2

(9)

Assume that such a solution z exists. Then multiplying (9) by ξ ∈ X , integrating in , we obtain    

(−z)ξ d x = −



zξ d x =



f2 ξ d x

By integration by parts and taking into account the boundary condition of the form ∂z − T z − f 1 = 0 on V , we obtain ∂ν 

 a(z, ξ) =



f2 ξ d x +

a(z, ξ) =



∇z∇ξ d x +

a(z, z) =

(10)

V

∇T z ∇T ξ d





Since

f 1 ξ d





where

V



|∇z|2 d x +

V

|∇T z|2 d

the sesquilinear form a is strongly coercive on X , and by the Lax-Milgram Lemma, problem (10) admits a unique solution z ∈ X . By taking a test function ϕ ∈ D(0, 1), it is easy to see that z satisfies system (9) in the distributional sense. This also shows that z belong to H 2 () because z = − f 2 ∈ L 2 () Setting ϑ = − f 1 we have shown that (z, ϑ) belongs to D(A) and is a solution of −AU = f . Therefore, we deduce that 0 ∈ ρ(A). Then by the resolvent identity, for

On the Polynomial Decay of the Wave Equation with Wentzell Conditions

213

λ > 0 small enough we have R(λ I − A) = H (see Theorem 1.2.4 in [24]). Using the Lumer-Phillips Theorem, the operator A generates a C0 -semigroup of contractions on H and thus completes the proof of the theorem.   In order to study the polynomial stabilisation of the aforementioned problem, we analyse the exponential stability of a one-dimensional problem using a Fourier method.

3 Exponential Stability of a One-Dimensional Problem Here, we proceed as in [26] assuming that the boundary V is composed of two parallel sides, i.e. V = {(1, y) : 0 < y < 1} ∪ {(0, y) : 0 < y < 1} = V1 ∪ V2 we consider the partial Fourier expansion of z z(x, y, t) =

∞ 

z ( jπ) (x, t) sin( jπ y)

j=1

The Fourier coefficients z ( jπ) of z satisfy the following problem with control acting in 1 and 0: ⎧ ) 2 (J ) ⎪ z tt(J ) − z (J =0 in (0, 1), ∀ t > 0, ⎪ x x + (J ) z ⎪ ⎨ (J ) 2 (J ) ∀ t > 0, z x (0, t) = (J ) z (0, t) + z t(J ) (0, t), (11) (J ) (J ) 2 (J ) ⎪ (1, t) = −(J ) z (1, t) − z (1, t), ∀ t > 0, z t ⎪ x ⎪ ⎩ (J ) (J ) (J ) (J ) z (·, 0) = z 0 , z t (·, 0) = z 1 . where J = jπ, z 0 (x, y) = sin( jπ y), Introduce the energy E J (t) =

1 2



1 0

∞

( jπ) j=1 z 0 (x) sin( jπ y)

and z 1 (x, y) =

(J )

) 2 2 (J ) (((z (J (x, t))2 + (z t (x, t))2 ) d x + x (x, t)) + J (z

∞

( jπ) j=1 z 1 (x)

J 2 (J ) ((z (1, t))2 + (z (J ) (0, t))2 ). 2

This system is exponentially stable according to V. Komornik [18] but the decay rate is related to J . So there are two positive constants C(J ) and ω(J ) such that E J (t) ≤ C(J )e−ω(J )t E J (0).

Lemma 1 Let z J be a regular solution of (11), then E J (t) = −((z t(J ) (1, t))2 + (z t(J ) (0, t))2 ) 

(12)

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Proof Let z (J ) be a regular solution of (11), integrating by parts yields 

E J (t)

 (J ) (J ) (J ) (J ) (J ) (J ) (J ) (J ) = 01 (∂x (z x z t ) − z x x z t + J 2 z t z (J ) + z tt z t )d x + J 2 (z t (1, t)z (J ) (1, t) (J )

+z t (0, t)z (J ) (0, t))  (J ) (J ) (J ) (J ) (J ) (J ) = 0 (∂x (z x z t )d x + 01 z t (−z x x + J 2 z (J ) + z t t)d x + J 2 (z t (1, t)z (J ) (1, t) 1

(J )

+z t (J )

(J )

= z x (1, t)z t

(J )

(J )

(1, t) − z x (0, t)z t

(0, t)z (J ) (0, t))

(J ) (J ) (0, t) + J 2 (z t (1, t)z (J ) (1, t) + z t (0, t)z (J ) (0, t))

Using the boundary conditions, we deduce that E J (t) = −((z t(J ) (1, t))2 + (z t(J ) (0, t))2 ) 

  In order to study the explicit dependence of the constants with respect to J, we write z (J ) as follows: z (J ) = f + g where f satisfies the problem (11) in the absence of dissipation, g is the remainder. They are, respectively, solutions of the two problems: ⎧ f tt − f x x + J 2 f = 0 in (0, 1), ∀t > 0 ⎪ ⎪ ⎨ f (0, t) = J 2 f (0, t), ∀t > 0 x 2 (1, t) = −J f (1, t), ∀t > 0 f ⎪ x ⎪ ⎩ f (·, 0) = z 0(J ) , f t (·, 0) = z 1(J )

(13)

⎧ gtt − gx x + J 2 g = 0 in (0, 1), ∀t > 0 ⎪ ⎪ ⎨ ∀t > 0 gx (0, t) = J 2 g(0, t) + z tJ (0, t), 2 J (1, t) = −J g(1, t) − z (1, t), ∀t >0 g ⎪ x t ⎪ ⎩ g(·, 0) = 0, gt (·, 0) = 0.

(14)

We show an observability estimate for the system (13) by applying the spectral theory combined with Ingham’s inequality and then by a perturbation argument, we find the requested observability estimate for the problem (14). We start by studying the problem (13), for that we introduce the operator A J of L 2 (0, 1) in itself with domain D(A J ) = {v ∈ H 2 (0, 1) : vx (0) = J 2 v(0) and vx (1) = −J 2 v(1)} defined by A J v = −vx x + J 2 v, ∀v ∈ D(A J ).

On the Polynomial Decay of the Wave Equation with Wentzell Conditions

215

Proposition 1 A J is self-adjoint positive operator of L 2 (0, 1) in itself. Furthermore, the resolvent Rλ (A J ) is compact for all λ ∈ ρ(A J ). Proof • We denote (.) the inner product of L 2 (0, 1). 

1

∀υ ∈ D(A J ), (A J (υ), υ) =

(−∂x x υ + J 2 υ)υ) d x  1  1 (−∂x x υυ) d x + J 2 υ2 d x = 0 0  1  1 = −υx (1)υ(1) + (∂x υ)2 d x + J 2 υ2 d x 0 0  1  1 (∂x υ)2 d x + J 2 υ2 d x ≥ 0 = J 2 υ(1)2 + 0

0

0

We define D(A∗J ) = {v ∈ J 2 (0, 1) : ∃ c > 0, ∀u ∈ D(A J ), |(v, A J u)| ≤ cu L 2 (0,1) } and

∀v ∈ D(A∗J ), ∀u ∈ D(A J ), (A∗J v, u) = (v, A J u).

First, we start by showing that D(A J ) ⊂ D(A∗J ) and A∗J/D(A J ) = A J . Let u, v ∈ D(A J ). 

1

(v, A J u) = 



0

J 2 uv d x

0 1

=

1

v(−∂x x u) d x +



1

∂x v∂x u d x − u x (1)v(1) + u x (0)v(0) +

0



= vx (1)u(1) − vx (0)u(0) +

J 2 uv d x

0 1

0

Since vx (1) = −J 2 v(1) and u x (1) = −J 2 u(1) vx (0) = J 2 v(0) and u x (0) = J 2 u(0) Then



1

(v, A J u) =

u(−∂x x v) + J 2 uv d x = (A J v, u)

0

We deduce that



u(−∂x x v) d x − u x (1)v(1) + u x (0)v(0)

v ∈ D(A∗J ) and A∗J v = A J v

0

1

J 2 uv d x

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Now, we consider the space H 1 (0, 1) with the inner product 

1

u, ξ 1 =

(uξ + u x ξx + J 2 uξ) d x + J 2 u(1)ξ(1),

0

which defines a norm equivalent to the usual norm. We deduce that, for all f ∈ L 2 (0, 1), there exists a unique u ∈ H 1 (0, 1) such that 

1

∀ξ ∈ H (0, 1), u, ξ 1 = 1

f ξ dx

0

Taking ξ ∈ D(0, 1), we obtain u x x = u + J 2u − f we have u ∈ H 2 (0, 1),

in



D (0, 1)

u + AJ u = f

(15)

and  1 0

(u x + u x ξx + J 2 uξ)d x + J 2 u(1)ξ(1) − J 2 u(0)ξ(0) =

 1 0

f ξ d x,

∀ξ ∈ H 1 (0, 1)

By integrating by parts we get  1 0

(u + J 2 u − u x x )ξ d x + u x ξ|10 + J 2 u(1)ξ(1) − J 2 u(0)ξ(0) =

 1 0

f ξ d x,

∀ξ ∈ H 1 (0, 1)

Since u + J 2 u − u x x = f , it remains that (u x (1) + J 2 u(1))ξ(1) − u x (0)ξ(0) = 0,

∀ξ ∈ H 1 (0, 1)

Which implies u x (0) = 0 and u x (1) = −J 2 u(1) Since u ∈ H 2 (0, 1) then u ∈ D(A J ). Now we consider v ∈ D(A∗J ), there exists w ∈ L 2 (0, 1) such that 

1

∀ξ ∈ D(A J ), 0

 A J ξv d x =

1

wξ d x

0

Let u be the solution of (15) with f = v + w. So w = −v + u + A J u and

On the Polynomial Decay of the Wave Equation with Wentzell Conditions



1



1

A J ξv d x =

0



1

(u − v)ξ d x +

0

 A J uξ d x =

0

hence

 ∀ξ ∈ D(A J ),

1

1

217

 (u − v)ξ d x +

0

1

u AJ ξ dx

0

(A J ξ + ξ)(v − u) d x = 0

0

By taking ξ solution of (15) with f = v − u, we obtain v ≡ u hence v ∈ D(A J ) and D(A∗J ) ⊂ D(A J ) Hence A J is a self-adjoint operator. • If λ = −1 − J 2 , then ∂u 2 ((λI − A J )u, u) = −u2 2 − J 2 u2 2 − J 2 u2 2 − − J 2 (u(1))2 , u ∈ D(A J )  L (0,1) L (0,1) L (0,1) ∂x L 2 (0,1) ≤ −(u2 2

L (0,1)

+

∂u 2 + J 2 (u(1))2 ), u ∈ D(A J )  ∂x L 2 (0,1)

We see that (λI − A J )u2L 2 (0,1) ≥ u2L 2 (0,1) + 

∂u 2  2 + J 2 (u(1))2 , u ∈ D(A J ) ∂x L (0,1)

This means that λ ∈ ρ(A J ) and Rλ (A J ) = (λI − A J )−1 is compact

 

The following theorem gives a characterisation on the spectrum of the operator A J . Theorem 2 The roots of the equation tan() =

2J 2 − J4

2

(16)

√ are the eigenvalues of A J noted λ2 with  = λ2 − J 2 . The normalised eigenvectors are given by k (17) ϕk (x) = αk (sin(k x) + 2 cos(k x)), ∀x ∈ (0, 1), J  where k = λ2k − J 2 and αk satisfy αk2 =

2k

2J 4 , + J 4 + 2J 2

√ √ 2 J2 2 αk k αk ∼ −→ 0 as k −→ ∞ and ≥ , ∀ k ∈ N, k J2 2J 2 In addition, we have

(18)

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K. Laoubi and D. Seba

|ϕk (0)|2 = |ϕk (1)|2 =

αk2 2k 22k = , ∀ k ∈ N, 2 J4 k + J 4 + 2J 2

|ϕk (0)| = |ϕk (1)| ≥

(19)

√ 2 , ∀ k ∈ N, 2J 2

(20)

Finally, we obtain the gap condition √ π 2 λk+1 − λk ≥ γ = , ∀ k ∈ N. 8J

(21)

Proof • Let ϑ ∈ D(A J ), using Green’s formula, we write 1 0

1 = 0 (−ϑx x + J 2 ϑ)ϑ(x)d x 1 = 0 (ϑx )2 + (−ϑx ϑ |10 ) + 0 J 2 ϑ2 (x)d x 1 = 0 ((ϑx )2 + J 2 ϑ2 (x))d x − ϑx (1)ϑ(1) + ϑx (0)ϑ(0)

(A J ϑ)(x)ϑ(x)d x

1

the boundary conditions of (13) show that 

1



1

(A J ϑ)(x)ϑ(x)d x =

0

 ((ϑx )2 + J 2 ϑ2 (x))d x + J 2 ϑ2 (1) + J 2 ϑ2 (0) ≥ J 2

0

1

(ϑ)2 d x

0

Consequently, the eigenvalues are greater than J 2 Now we assume that λ2 ≥ J 2 . We shall consider the following system: ⎧ ⎨ −ϕx x + J 2 ϕ = λ2 ϕ in (0, 1), ϕx (1) = −J 2 ϕ(1) ⎩ ϕx (0) = J 2 ϕ(0) The first equation of (22) shows that there exist α, β ∈ R such that ϕ(x) = α sin(x) + β cos(x) with  =

√

λ2 − J 2 . As ϕx (x) = α cos(x) − β sin(x)

the second boundary condition gives ϕx (0) = J 2 β = α ⇒ β = Then we can write

α J2

   ϕ(x) = α sin( x) + 2 cos( x) , J

(22)

On the Polynomial Decay of the Wave Equation with Wentzell Conditions

219

If  = 0 then ϕ = 0, which is impossible. Hence λ2 = J 2 cannot be an eigenvalue of A J . • From the condition ϕx (1) = −J 2 ϕ(1), we obtain α cos() − β sin() = −J 2 (α sin  + βcos) hence

sin  α + J 2 β = = tan  2 α − J α cos 

Using the fact that β =

α , J2

we have tan() =

2J 2 2 − J 4

The roots of this equation are simple and verify kπ < k < kπ + π2 , kπ − π2 < k < kπ, 2

if if

k > J 2 , 0 < k < J 2 .

√

• Let’s show that αk ∼ J k 2 −→ 0 when k −→ ∞ From the normalisation hypothesis  0

1

ϕ2k (x)d x = 1

we obtain 

1 0

αk2 [sin2 (k x) +

2k k cos2 (k x) + 2 (sin(2k x))]d x = 1 4 J J

Using the trigonometric formulas sin2 (k x) =

1 − cos(2k x) 2

cos2 (k x) =

1 + cos(2k x) 2

and

we obtain  1 2 2 1 cos(2k x) k 2 + k4 + k4 cos(2k x) + 2 sin(2k x))d x = 1 αk ( − 2 2 2J 2J J 0 This identity is equivalent to

(23)

220

K. Laoubi and D. Seba

αk2



+

1 2

=

2k 2J 4

αk2 2k 2J 4



2

+

1 2J 2

+ ( 4kkJ 4 −

J4 2k

+1+

J2 2k

1 ) sin(2k ) 4k

−

2J 4 sin(2k ) ) 4k 2k

−

+ (2 −

k 2k J 2 J2 2k

cos(2k )  cos(2k )



= 1. Knowing that | sin(k )| = sin(2k ) =

2k J 2 , | cos(k )| 2k +J 4 4k J 2 (2k −J 4 )

(2k +J 4 )2

,

we obtain αk2 =

2 −J 4

= | k2 +J 4 |, k

tan(2k ) =

cos(2k ) = 4k J 2 (2k −J 4 ) , 4k +J 8 −62k J 4

4k +J 8 −62k J 4

(2k +J 4 )2

2J 4 2k + J 4 + 2J 2

, (24)

(25)

since k −→ ∞ as k −→ ∞, then αk2 2k ∼ 1, 2J 4 which implies αk ∼

√ J2 2 −→ 0 since k −→ ∞. k

Since k ≥ 1 and J ≥ 1, we may write αk2 2k (42k − 1)J 4 − (2k + 2J 2 ) N 1 = − = 2 4 4 4 4 2 J 2J D 2J (k + J + 2J ) N ≥ (42k − 1)J 2 − (2k + 2J 2 ) = 2k (4J 2 − 1) − 3J 2 ≥ 4J 2 − 1 − 3J 2 = J 2 − 1 ≥ 0

This implies that

√ 2 αk k ≥ , ∀ k ∈ N, 2 J 2J 2

• We have ϕk (1) = αk sin(k ) + αJk 2 k cos(k ), = cos k (αk tan k + αJk 2 k ) ϕk (0) = Using (24) and (25), we obtain

αk k J2

On the Polynomial Decay of the Wave Equation with Wentzell Conditions

αk2 2k J4

|ϕk (0)|2 = |ϕk (1)|2 =

=

2k

221

22k . + J 4 + 2J 2

Since k ≥ 1 and J 2 ≥ 1 we deduce that √ 2 . |ϕk (0)| = |ϕk (1)| ≥ 2J 2 • From k =



λ2k − J 2 , we get λk+1 − λk

=



2k+1 + J 2 −

=√



2k+1 −2k 2 k+1 +J 2 +

√

2k + J 2

2k +J 2

k+1 +k

√

= (k+1 − k ) √

2k+1 +J 2 +

Using (23), we obtain k+1 − k ≥

2k +J 2

.

π 2

(26)

From (26), we write λk+1 − λk ≥

π 2

√

k+1 +k

√

2k+1 +J 2 +

≥

2k +J 2

π 2 1+

=

π 2 1+

1  J2 2 k+1

+

2k 2 k+1

+



1+  k k+1 J2 2k+1

J2 2k+1

+

2k 2k+1

+

J2 2k+1

.

Since k+1 > k , we obtain λk+1 − λk ≥

1 π  4 1+

J2

=

2k+1

π  4J

1 1 J2

+

1 2k+1

This implies that λk+1 − λk ≥

π √ , since k ≥ 1 and J ≥ 1 4J 2

which proves (21). We are able to give an estimate on the energy of f .

 

222

K. Laoubi and D. Seba

Proposition 2 Let E f (t) be the energy of f solution of (13) defined by E f (t) =

1 2



1 0

(( f x2 (x, t)) + J 2 ( f 2 (x, t)) + ( f t2 (x, t))2 ) d x +

J2 (( f 2 (1, t)) + ( f 2 (0, t))). 2



with E f (t) = 0 Then there exist a positive constant C independent on J such that for all T > C J , we have 

T

E f (0) ≤ C 0

(( f t2 (1, t)) + ( f t2 (0, t))) dt,

√ ∀ T ≥ 16J 2.

(27)

Proof The spectral theory allows to write f (x, t) =



( f 0k cos(tλk ) + f 1k

k≥0

sin(tλk ) )ϕk (x) λk

where f 0k (resp. f 1k ) are the Fourier coefficients of z 0(J ) (resp.z 1(J ) ), i.e.:   z 0(J ) = k≥0 f 0k ϕk and z 1(J ) = k≥0 f 1k ϕk This implies that f t (1, t) =

 cos(tλk ) (− f 0k sin(tλk ) + f 1k )λk ϕk (1) λk k≥0

f t (0, t) =

 cos(tλk ) (− f 0k sin(tλk ) + f 1k )λk ϕk (0) λk k≥0

and

Then according to the gap condition (21) and using Ingham’s inequality [16] (see also Lemma 3.3 [26]), we get that  4π  ∞ γ C1  2 (λk | f 0k |2 + | f 1k |2 )|ϕk (1)|2 ≤ | f t (1, t)|2 dt, γ k=0 0 and

 4π  ∞ γ C2  2 (λk | f 0k |2 + | f 1k |2 )|ϕk (0)|2 ≤ | f t (0, t)|2 dt, γ k=0 0 



for C1 > 0 and C2 > 0. using inequality |ϕk (1)|2 = |ϕk (0)|2 ≥ ∞  k=0

1 , 2J 4

we deduce that 

(λ2k | f 0k |2 + | f 1k |2 ) ≤ C3 J 3

0

4π γ

| f t (1, t)|2 dt,

On the Polynomial Decay of the Wave Equation with Wentzell Conditions

and

 ∞  2 2 2 3 (λk | f 0k | + | f 1k | ) ≤ C3 J k=0

4π γ

223

| f t (0, t)|2 dt,

0

Which implies  4π ∞  γ C3 3 2 2 2 J (λk | f 0k | + | f 1k | ) ≤ (| f t (1, t)|2 + | f t (0, t)|2 ) dt, 2 0 k=0 where C3 is a positive constant independent of J . We now conclude by the identity (cf [18, 22]) that ∞  (λ2k | f 0k |2 + | f 1k |2 ) = z 0(J ) 2D(A1/2 ) + z 1(J ) 2 J

k=0

where · is the L 2 (0, 1) norm, and the property  1 1/2 2 (vx (x)2 + J 2 v(x)2 ) d x + J 2 (v(1)2 + v(0)2 ), ∀ v ∈ D(A J ) 1/2 = A J v = (A J v, v) = D(A J ) 0

v2

where (·, ·) is the L 2 (0, 1) product. Since z 0(J ) = f 0 (., 0), z 1(J ) = f t (., 0) then ∞ 

(λ2k | f 0k |2 + | f 1k |2 ) =

1 0

( f x (x, 0)2 + J 2 f (x, 0)2 ) + f t (x, 0)2 d x + J 2 ( f (1, 0)2 + f (0, 0)2 )

k=0

= 2E f (0),

Hence, there is a constant C4 > 0, independent of J , such that  E f (0) ≤ C4 J

3

4π γ

(( f t (1, t))2 + ( f t (0, t))2 ) dt

0

for all T >

4π γ

√ = 16J 2 we get  E f (0) ≤ C

T

(( f t (1, t))2 + ( f t (0, t))2 ) dt

0

  The following proposition gives an estimate on g. Proposition 3 There exists a constant C > 0 and T > 0 such that the solution g of (14) satisfies

224



K. Laoubi and D. Seba T

0

 (gt2 (1, t)

+

gt2 (0, t)) dt

T

≤ C(T + T + 1) 2

((z t(J ) (1, t))2 + (z t(J ) (0, t))2 )dt

0

(28)

We write the solution g of problem (14) in the form g = p + q, where p (resp.q) is solution of the same problem as g but with dissipation in 1 only ( resp. 0 only). Then ⎧ 2 in (0, 1), ∀t > 0, ⎪ ⎪ ptt − px x + 2J p = 0, ⎨ ∀t > 0, px (0, t) = J p(0, t) (29) 2 (1, t) = −J p(1, t) − k (t), ∀t > 0, p ⎪ x 1 ⎪ ⎩ p(·, 0) = 0, pt (·, 0) = 0. ⎧ 2 in (0, 1), ∀t > 0, ⎪ ⎪ qtt − qx x + J2 q = 0, ⎨ qx (0, t) = J q(0, t) + k2 (t), ∀ t > 0, ∀t > 0, qx (1, t) = −J 2 q(1, t), ⎪ ⎪ ⎩ q(·, 0) = 0, qt (·, 0) = 0,

and

(30)

where



z t(J ) (0, t) if 0 < t < T 0 else. (31) The proof of Proposition (3) is based on the two following lemmas: k1 (t) =

z t(J ) (1, t) if 0 < t < T 0 elsewhere,

and k2 (t) =

Lemma 2 There exists C > 0 and T > 0 such that 

T 0

 ( pt2 (1, t) + pt2 (0, t)) dt ≤ C(T 2 + T + 1)

T

(z t(J ) (1, t))2 dt

(32)

0

Proof Using the multiplier technique [18], we multiply the differential equation in (29) by (x − 21 )(2T − t) px . By integration by parts in Q = (0, 1) × (0, 2T ), we get for regular solutions 

 1 1 2T t=2T 0= [(x − )(2T − t) pt px ]t=0 d x − (2T − t)[(x − 1/2) pt2 ]x=1 x=0 dt 2 2 0 0   1 + (x − 1/2) pt px d xdt + (2T − t)[ pt2 + px2 − J 2 p 2 ] dtd x 2 Q Q  1 2T − [(x − 1/2)( px2 − J 2 p 2 )]x=1 x=0 (2T − t) dt. 2 0 1

This identity is equivalent to

On the Polynomial Decay of the Wave Equation with Wentzell Conditions

 0=

Q

225

 1 (2T − t)[ pt2 + px2 − J 2 p2 ] dtd x 2 Q  2T  1 2T (2T − t)( pt2 (1, t) + pt2 (0, t))dt + (2T − t)[J 2 p2 (1, t) − px2 (1, t)] dt 4 0 0  2T (2T − t)[J 2 p2 (0, t) − px2 (0, t)] dt

(x − 1/2) pt px d xdt +

−

1 4

+

1 4 0

which allows to write   1 2T (2T − t)( pt2 (1, t) + pt2 (0, t))dt = (x − 1/2) pt px d xdt 4 0 Q  1 (2T − t)[ pt2 + px2 − J 2 p2 ] dtd x + 2 Q  1 2T (2T − t)[J 2 p2 (1, t) − px2 (1, t)] dt + 4 0  1 2T + (2T − t)[J 2 p2 (0, t) − px2 (0, t)] dt. 4 0

We recall that px (0, t) = J 2 p(0, t), then J 2 p 2 (0, t) − px2 (0, t) = J 2 p 2 (0, t) − J 4 p 2 (0, t) = (J 2 − J 4 ) p 2 (0, t) ≤ 0, ∀J ≥ 2. Then using the Cauchy-Schwartz inequality we deduce that 

 (x − 1/2) pt px d xdt ≤ C Q

Q

( pt2 + px2 ) dtd x,

which gives   1 2T (2T − t)( pt2 (1, t) + pt2 (0, t))dt ≤ C(T + 1) ( pt2 + px2 ) dtd x (33) 4 0 Q  1 2T + (2T − t)[J 2 p2 (1, t) − px2 (1, t)] dt, 4 0

Using the fact that px (1, t) = −J 2 p(1, t) − k1 (t), we obtain J 2 p 2 (1, t) − px2 (1, t) = J 2 p 2 (1, t) − (J 2 p(1, t) + k1 (t))2 = (J 2 − J 4 ) p 2 (1, t) − 2J 2 p(1, t)k1 (t) − k12 (t) For ε > 0, one has

226

K. Laoubi and D. Seba

1 |2J 2 p(1, t)k1 (t)| ≤ εJ 4 p 2 (1, t) + k12 (t)  which implies 1 J 2 p(1, t)2 − px (1, t)2 ≤ (J 2 − J 4 + J 4 ) p(1, t)2 + ( − 1)k1 (t)2 .  We then fix ε > 0 such that J 2 − J 4 + J 4 ≤ 0, ∀J ≥ 2. we get J 2 p(1, t)2 − px (1, t)2 ≤ Ck1 (t)2 .

(34)

By substituting the estimate (34) into (33), we obtain 1 4



2T 0

 (2T − t)( pt2 (1, t) + pt2 (0, t))dt ≤C(T + 1) 

2T

+C 0

Q

( pt2 + px2 ) dtd x

(2T − t)k12 (t) dt

(35)

Now we define the energy of the solution p by E p (t) =

 1 1 2 J2 (( p2 (1, t)) + ( p2 (0, t))). (( px (x, t)) + J 2 ( p 2 (x, t)) + ( pt2 (x, t))) d x + 2 0 2

Integrating by parts for regular solutions, we obtain 

E p (t) =

 1 0

pt ( ptt − px x + J 2 p) d x + [ px pt ]10 + J 2 ( pt (1, t) p(1, t) + pt (0, t) p(0, t))

= px (1, t) pt (1, t) − px (0, t) pt (0, t) + J 2 ( pt (1, t) p(1, t) + pt (0, t) p(0, t))

Using the boundary conditions of the problem (29), we obtain 

E p (t) = (−J 2 p(1, t) − k1 (t)) pt (1, t) − J 2 p(0, t) pt (0, t) + J 2 pt (1, t) p(1, t) + J 2 pt (0, t) p(0, t) = −k1 (t) pt (1, t). Hence



E p (t) = −k1 (t) pt (1, t)

(36)

Integrating the last result between 0 and s and reminding that E p (0) = 0 we get

On the Polynomial Decay of the Wave Equation with Wentzell Conditions



S

E p (S) − E p (0) = 0





E p (t) dt = −

S

227

k1 (t) pt (1, t) dt.

0

Integrating between 0 and 2T , and using Fubini’s theorem, we obtain 



2T 0



2T

E p (S)ds = − 0

S



2T

pt (1, t)k1 (t)dt ds = −

0

(2T − t) pt (1, t)k1 (t)dt.

0

For  > 0, one has 

2T



2T

E p (S)ds ≤ 

0

0

We also have

pt2 (1, t)(2T − t) dt + 

 Q

( pt2 + px2 ) dtd x ≤ 2

1 4

2T



2T

0

k12 (t)(2T − t)dt.

(37)

E p (S)ds

0

Substituting this last inequality into (35), we get 1 4



2T 0

 (2T − t)( pt2 (1, t) + pt2 (0, t))dt ≤ C(T + 1)

2T



2T

E p (S)ds +

0

0

k12 (t)(2T − t)dt.

Using (36), we obtain  2T  1 2T (2T − t)( pt2 (1, t) + pt2 (0, t))dt ≤ C(T + 1) pt2 (1, t)(2T − t) dt 4 0 0  2T 1 k12 (t)(2T − t)dt + C(T + 1) 4 0  2T k12 (t)(2T − t)dt. + 0

Choosing  such that C(T + 1) = 18 , we obtain  2T  1 2T (2T − t)( pt2 (1, t) + pt2 (0, t))dt ≤ C(T + 1)2 k12 (t)(2T − t)dt 8 0 0  2T k12 (t)(2T − t)dt. + 0

As T ≤ 2T − t ≤ 2T on [0, T ] and using the definition of k1 (t), we get the desired result, i.e.  T  T 2 2 2 ( pt (1, t) + pt (0, t)) dt ≤ C(T + T + 1) (z t(J ) (1, t))2 dt, 0

for a positive constant C.

0

 

228

K. Laoubi and D. Seba

Lemma 3 There exists a positive constant C such that for all T > 0 the solution q of (30) satisfies 

T 0

 (qt2 (1, t)

+

qt2 (0, t)) dt

T

≤ C(T + T + 1) 2

(z t(J ) (0, t))2 dt

(38)

0

 

Proof It is sufficient to repeat the proof of Lemma 3.5. Let’s go back to the proof of Proposition 3 Proof By combining the estimates (32) and (38), we get  T 0

 ( pt2 (1, t) + qt2 (1, t) + pt2 (0, t) + qt2 (0, t)) dt ≤ C (T 2 + T + 1)

 T 0

(J )

((z t

(J )

(1, t))2 + (z t

(0, t))2 ) dt

Since gt2 (1, t) = ( pt (1, t) + qt (1, t))2 ≤ 2{ pt2 (1, t) + qt2 (1, t)} and gt2 (0, t) = ( pt (0, t) + qt (0, t))2 ≤ 2{ pt2 (0, t) + qt2 (0, t)} then  0

T

 (gt2 (1, t) + gt2 (0, t)) dt ≤ C(T 2 + T + 1) 

T 0

z t2 (0, t) + z t2 (1, t) dt,



where C = 2C , C > 0

 

We state the following result about the exponential stability: Theorem 3 There exist two constants C1 > 0, C2 > 0 such that C2

E J (t) ≤ C1 e− J 3 t E J (0),

∀t ≥ 0

(39)

Proof By the splitting z (J ) = f + g, we see that E J (0) = E f (0) and by Proposition 2 we get 

T

E f (0) = E J (0) ≤ C

( f t (1, t)2 + f t (0, t)2 ) dt,

0

√

for T > 16J 2 . Since

and

f t (1, t) = z t(J ) (1, t) − gt (1, t) f t (0, t) = z t(J ) (0, t) − gt (0, t)

On the Polynomial Decay of the Wave Equation with Wentzell Conditions

229

Then f t2 (1, t) ≤ 2{z t2 (1, t) + gt2 (1, t)} f t2 (0, t) ≤ 2{z t2 (0, t) + gt2 (0, t)}. Using Proposition 3, we obtain 

T

E J (0) ≤ C(T + T + 1) 2

0

(z t2 (1, t) + z t2 (0, t)) dt,

√ √ for T > 16J 2 , choosing TJ = (16J 2 + 1), we arrive at 

TJ

E J (0) ≤ C(J 2 + J + 1) 0

(z t2 (1, t) + z t2 (0, t)) dt.

Recalling Lemma 1, we then conclude that E J (TL ) ≤ E J (0) ≤ C(J 2 + J + 1)(E J (0) − E J (TJ )), or E J (TJ )(1 + C(J 2 + J + 1)) ≤ C(J 2 + J + 1)E J (0) which finally gives E J (TJ ) ≤ γ J E J (0), where γ J =

C(J 2 +J +1) . 1+C(J 2 +J +1)

Theorem 3.3 of [28] leads to E J (t) ≤

with ω J =

1 TJ

ln

1 γJ

∼

C J3

1 −ω J t e E J (0), γJ

and γ J ∼ 1 −

C J2

 

Remark 1 The result of exponential stability obtained in this section for the onedimensional problem (11) allows us to deduce in the next section, the polynomial stability of the Wentzell problem (1) using a Fourier analysis (cf. [26, 31]).

4 Polynomial Stability In this section, we prove the polynomial stability of the problem (1) in the case when V is defined by V = {(1, y) : 0 < y < 1} ∪ {(0, y) : 0 < y < 1},

230

K. Laoubi and D. Seba

we recall the partial Fourier expansion of z z(x, y, t) =

∞ 

z ( jπ) (x, t) sin( jπ y)

j=1

The Fourier coefficients z ( jπ) of z satisfy the problem (11) with J = jπ. The associated energy defined by 1 E jπ (t) = 2

 0

1



( jπ)

(z (x jπ) (x, t))2 + j 2 π 2 (z ( jπ) (x, t))2 + (z t

 (x, t))2 d x +

j 2 π 2  ( jπ) z (1, t)2 + z ( jπ) (0, t)2 , 2 we have

∞

E(t) =

1 E jπ (t). 2 j=1

By Theorem 3 and Fourier analysis, we obtain the following polynomial stability for the problem (1). Theorem 4 For all k ∈ N , there exists a constant Ck > 0 such that for all (z 0 , z 1 ) ∈ 3 3 3 j H 2 k+1 () × H 2 k () with z 0| j ∈ H 2 k+1 (1 ), j = 1, 2, the solution z of (1) sat1 isfies ∞ Ck  3k j E jπ (0), ∀ t > 0, (40) E(t) ≤ k t j=1 where

∞ 

j 3k E jπ (0) ≤ z 0 2

3

H 2 k+1 ()

j=1

+ z 0|1 2

3

H 2 k+1 (1 )

+ z 1 2

3

H 2 k ()

.

is a higher order energy. Proof For a fixed k ∈ N , there exists a positive constant ck > 0 such that x k e−x ≤ ck , ∀x ≥ 0. This estimate in (39) leads to E jπ (t) ≤ C1

ck j 3k Ck E jπ (0) = k j 3k E jπ (0), ∀ t > 0, k k t C2 t

By Fourier analysis, we finally obtain

On the Polynomial Decay of the Wave Equation with Wentzell Conditions

E(t) =

∞ 

E jπ (t) ≤

j=1

231

∞ Ck  3k j E jπ (0), ∀ t > 0. t k j=1

By Parseval’s inequality and using the results given in [18], it is clear that ∞ 

j 3k E jπ (0) ≤ z 0 2

j=1

3

H 2 k+1 ()

+ z 0|1 2

3

H 2 k+1 (1 )

+ z 1 2

3

H 2 k ()

,  

5 Conclusion In this work, we have shown the polynomial decay of the Wentzell problem for the wave equation in the general case where the boundary V is composed of two parallel parts, i.e. V = {(1, y) : 0 < y < 1} ∪ {(0, y) : 0 < y < 1}. We obtained a result analogous to that proved in the particular case [31] where the subset V = {(1, y) : 0 < y < 1} does not satisfy the geometric condition of the control but with less regular initial data. The Fourier analysis used in the third section allows us to obtain the explicit dependence of the constants with respect to J and also to specify the domain of the initial data verifying the estimation of the polynomial stability.

References 1. Akil, M., Wehbe, A.: Stabilization of multidimensional wave equation with locally boundary fractional dissipation law under geometric conditions. Mathematical Control & Related Fields 9(1), 97–116 (2019) 2. Ammari, K., Gerbi, S.: Interior feedback stabilisation of wave equations with dynamic boundary delay. Zeitchrift Fur Analysis And Ihre AnwendunGen. 36(03), 297–327 (2017) 3. Ammari, K., Liu, Z., Tucsnak, M.: Decay rates for a beam with pointiwse force and moment feedback. Math. Control Signals system. 15(3), 229–255 (2002) 4. Bassam, M., Mercier, D., Nicaise, S., Wehbe, A.: Polynomial stability of the Timoshenko system by one boundary damping. J. Math. Anal. Appl. 425(2), 1177–1203 (2015) 5. Beniani, A., Benaissa, A., Zennir, K.: Polynomial Decay of Solutions to the Cauchy Problem for a Petrovsky-Petrovsky System-in Rn . Acta Appl Math 146, 67–79 (2016) 6. Borichev A, Tomilov Y (2010) Optimal polynomial decay of functions and operator semigroups. Mathematical Reviews (MathSciNet): MR2011c:47091 Math Ann 347(2): 455–478 7. Brezis, H.: Opérateurs maximaux monotones et semi-groupes de contraction dans les espaces de Hilbert. North Holland, Amsterdam (1973) 8. Buffe, R.: Stabilization of the wave equation with Ventcel boundary conditions. J. Math. Pures Anal. 108, 207–259 (2017) 9. Duyckaerts, T.: Optimal decay rates of the energy of a hyperbolic-parabolic system coupled by an interface. Asymptot. Anal. 51, 17–45 (2007) 10. Feller, W.: Generalized second order differential operators and their lateral conditions. Illinois J. Math. 1, 459–504 (1957)

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11. Feller, W.: The parabolic differential equations and the associated semi-groups of transformations. Ann. Math. 2(55), 468–519 (1952) 12. Grisvard, P.: Elliptic problems in nonsmooth domains. Monographs and Studies in Mathematics, vol. 24 Pitman, Boston-London-Melbourne (1985) 13. Guesmia, A.: Asymptotic behavior for coupled abstract evolution equations with one infinite memory. Appl. Anal. 94(1), 184–217 (2015) 14. Heminna, A.: Stabilisation de problemes de Ventcel. C. R. Acad. Sci. Paris 328, série I 1171– 1174 (1999) 15. Heminna, A.: Stabilisation frontière de problèmes de Ventcel. ESAIM Control Optim. Calc. Var. 5, 591–622 (2000) 16. Ingham, A.E.: Some trigonometrical inequalities with applications in the theory of series. Math. Z. 41, 367–369 (1936) 17. Jaffard, S., Tucsnak, M., Zuazua, E.: Singular inter stabilization of the wave equation. J. Diff. Equa. 145, 184–215 (1998) 18. Komornik, V.: Exact controllability and stabilization, the multiplier method, vol. 36 of RMA. Masson, Paris (1994) 19. Lemrabet, K., Bendali, A.: The effect of a thin coating on the scattering of a time-harmonic wave for the Helmholtz equation. SIAM J. Appl. Math. 56(6), 1664–1693 (1996) 20. Lemrabet, K.: Le problème de Ventcel pour le système de l’élasticité dans un domaine de R3 . C. R. Acad. Sci. Paris Sér. I Math. 304(6), 151–154 (1987) 21. Littman, W., Liu, B.: On the spectral properties and stabilization of acoustic flow. SIAM J. Appl. Math. 59(1), 17–34 (1999) 22. Lions, J., Magenes, E.: Problèmes aux limites non homogènes et applications. I-III Dunod Paris 1968–1970 Masson, Paris (1994) 23. Liu, Z., Rao, B.: Characterization of polynomial decay rate for the solution of linear evolution equation. Z. Angew. Math. Phys. 56(4), 630–644 (2005) 24. Liu, Z., Zheng, S.: Semigroups Associated with Dissipative Systems. Research Notes in Mathematics, vol. 398, Chapman Hall/CRC (1999) 25. Muñoz Rivera, J., Qin, Y.: Polynomial decay for the energy with an acoustic boundary condition. Appl. Math. Lett. 16(2), 249–256 (2003) 26. Nicaise, S., Laoubi, K.: Polynomial stabilization of the wave equation with Ventcel’s boundary conditions. Math. Nachr. 283(10), 1428–1438 (2010) 27. Nicaise, S., Li, H., Mazzucato, A.: Regularity and a priori error analysis of a Ventcel problem in polyhedral domains. [math.AP] (2016). arXiv: 1606.04423v1 28. Nicaise, S.: Stability and controllability of an abstract evolution equation of hyperbolic type and concrete applications. Rend. di Mat. Ser. VII 23, 83–116 (2003) 29. Rao, B.: Stabilization of elastic plates with dynamical boundary control. SIAM J. Control Optim. 36(1), 148–163 (1998) 30. Ventcel, A.D.: On boundary conditions for multi-dimensional diffusion processes. Theor. Probab. Appl. 4, 164–177 (1959) 31. Zhang, X., Zuazua, E.: Decay of the solutions of the system of thermoelasticity of type III. Commun. Contemp. Math. 5, 25–83 (2003) 32. Zhang, X., Zuazua, E.: Long-time behavior of a coupled heat-wave system arising in fluidstructure interaction. Arch. Ration. Mech. Anal. 184, 49–120 (2007) 33. Zhang, X., Zuazua, E.: Polynomial decay and control of a 1-D hyperbolic-parabolic coupled system. J. Differ. Equ. 204, 380–438 (2004)

Solutions of Fractional Verhulst Model by Modified Analytical and Numerical Approaches Shatha Hasan, Samir Hadid, Mohammed Al-Smadi, Omar Abu Arqub, and Shaher Momani

Abstract In this chapter, we are interested in the fractional release of the Verhulst model according to Caputo’s sense, which is popular in applying environmental, biological, chemical and social studies describing the population growth model. Such a model, which is sometimes called logistic growth model related to systems in which the rate of change depends on their previous memory. In the light of this, three advanced numerical and analytical algorithms are presented to obtain approximate solutions for different classes of logistical growth problems, including reproducing kernel algorithm, fractional residual series algorithm and successive substitutions algorithm. The first technique relies on the reproducing property that characterises a specific function of building a complete orthogonal system at desired Hilbert spaces. The RPS technique relies on residual error function and generalised Taylor series to reduce residual errors and generate a converging power series, while the last technique converts the fractional logistic model to Volterra integral equation based on Riemann-Liouville integral operator. To demonstrate consistency with the theoretical framework, some realistic applications are tested to show the accuracy and efficiency of the proposed schemes. Numerical results are displayed in tables and figures for different fractional orders to illustrate the effect of the fractional parameter on population growth behaviour. The results confirm that the proposed schemes are very convenient, effective and do not require long-term calculations.

S. Hasan · M. Al-Smadi (B) Department of Applied Science, Ajloun College, Al-Balqa Applied University, Ajloun 26816, Jordan e-mail: [email protected] S. Hadid · S. Momani Department of Mathematics and Sciences, College of Humanities and Sciences, Ajman University, Ajman, UAE O. Abu Arqub Faculty of Science, Department of Mathematics, Al-Balqa Applied University, Salt 19117, Jordan © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 D. Zeidan et al. (eds.), Computational Mathematics and Applications, Forum for Interdisciplinary Mathematics, https://doi.org/10.1007/978-981-15-8498-5_11

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1 Introduction Mathematical modelling is an important branch of mathematics to deal with various aspects of real applications that occur in engineering, science and social studies. There are many types of models, in particular, the mathematical model concerned with population dynamics, which is called the logistic growth model, first proposed by Pierre Verhulst in 1838 [1]. So, it is sometimes called the Verhulst model. Anyhow, the continuous form of the non-linear logistical equation proposed at that time was as follows:   W dW = CW 1 − , dt k where W (t) is population at time t, C > 0 is Malthusian parameter concerned with the growth rate and k describes the carrying capacity. Thus, if we set ω(t) = Wk , the following standard logistic differential equation can be converted as dω = δω(1 − ω), dt

(1)

which has the following exact solution ω(t) =

ω0 , ω0 + (1 − ω0 )e−δt

(2)

whereas ω0 = ω(0) is related to the initial population. Logistic differential equation has many applications such as modelling of population growth of tumours in medicine [2], modelling of social dynamics of replacement technologies by Fisher and Pry [3] and the adaptability of society to innovation. On the other hand, the generalisation of the classical concept of the derivatives of the integerorder system to derivatives of arbitrary order is nowadays a growing interest area of mathematics to study various natural phenomena. Recent advancement on fractional operators has demonstrated that generalised fractional models are much better compared to classic integer-order models. This feature is due to a variety of choices of the order that often lead to better results. The classical derivative describes the instantaneous rate of change of a given function, while the fractional parameter can be interpreted as a memory indication of the function variance considering previous moments. These benefits encouraged researchers to use fractional operators when studying complex physical phenomena, especially dealing with memory processes or viscoelastic materials [4–7]. To read more about fractional calculus applications in rheology modelling, fluid mechanics, biology, entropy theory, etc., the reader can refer to [8–11]. In fact, there are many definitions of fractional derivatives and integrals. Some of these are Riemann-Liouville, Caputo, Grunwald-Letinkov, Weyl, Riesz, Feller, Caputo-Fabrizio and Atangana-Baleanu [12–15]. In this chapter, the

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fractional derivative Dα is considered in the Caputo sense which has the property that the derivative of any constant is zero and hence the initial conditions of the fractional differential equation takes on the classical form similar to those of integer order. The fractional form of the logistic differential equation can be obtained by replacing the first-order derivative in Eq. (1) with a fractional derivative Dα , 0 < α ≤ 1. Consequently, we consider the fractional logistic differential equation (FLDE) of the following form: Dα ω(t) = δω(t)(1 − ω(t)), t > 0, δ > 0,

(3)

subject to the initial condition ω(0) = ω0 , ω0 > 0.

(4)

The stability, existence and uniqueness of the FLDE have been studied in [14]. For α = 1, the FLDE in (3) reduces to the classic LDE in (1). Recently, West in [15] has provided the function ω(t) =

∞  ω0 − 1 k=0

ω0

Eα (−kδt α ), t ≥ 0,

(5)

is the Mittag-Leffler funcas an exact solution to the FLDEs (3) and (4). Here, Eα (z) zn tion that is defined as a power series given by Eα (z) = ∞ n=0 (nα+1) . But in [16], Area et al. proved that the function in (5) is the solution of (3) if and only if the power is first order, so it is not the exact solution of arbitrary fractional order. Indeed, most fractional derivatives have a non-local property. So, the exact solutions of fractional differential equations are not easy to obtain. This is one of the reasons that why fractional calculus is more popular and a rich field for research. However, despite recent major efforts on the topic of fractional calculus, there are no methods in literature to produce exact solutions of non-linear fractional differential equations till now. So, approximate and numerical methods are needed. Several numerical techniques have been applied to solve fractional logistic problem in Caputo sense, including homotopy perturbation method spectral Laguerre collocation method, variational iteration method, operational matrices of Bernstein polynomials and finite difference method [17–23]. Different types of numerical method can be found in [24–31]. In this chapter, we employ three different methods to obtain approximate solutions for the FLDE and to study the effect of a fractional operator to the curves of population growth of species. The first method is the reproducing kernel Hilbert space (RKHS) method [32–35]. This method has many advantages; first, it is global in nature due to its ability to solve different mathematical and physical problems, especially non-linear problems; second, it is accurate and its numerical results can be obtained easily; third, it doesn’t require discretisation of the variables, and it is not affected by round off errors of computations. Reproducing kernel theory has important applications in mathematics, image processing, machine learning, finance and

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probability [36–40]. Recently, a lot of research work has been devoted to the applications of RKHS method for wide classes of stochastic and deterministic problems involving operator equations, differential equations and integro-differential equations. To see these applications, the reader can refer to [41–46] and the references therein. The second proposed method in this chapter is the fractional residual power series (FRPS) method in which the solution is assumed to have a generalised fractional Taylor series representation. This technique has been first developed to approximate numerical solutions for certain class of differential equations under uncertainty. Later, the generalised Lane-Emden equation has been investigated numerically by utilising the FRPS method. Also, the method was applied successfully in solving composite and non-composite fractional differential equations, fractional boundary value problems, stiff systems and time-fractional Fokker–Planck models [47–52]. Further, [53] asserts that the FRPS method is easy and powerful to construct power series solution for strongly linear and non-linear equations without terms of perturbation, discretisation and linearisation. Unlike the classical power series method, the FRPS method distinguishes itself in several important aspects such as it does not require making a comparison between the coefficients of corresponding terms and a recursion relation is not needed. It provides a direct way to ensure the rate of convergence for series solution by minimising the residual error-related [54]. In fact, it is just a repetition of Caputo derivative to obtain the values of unknown coefficients of desired fractional series solution by solving sequence of algebraic equations with choosing a fit initial data. The gained series solution and all fractional derivatives are valid for all mesh points of the domain of interest. The third technique that we apply in this chapter for solving FLDE depends on a property of Riemann-Liouville integral and Caputo derivative. We just apply the Riemann-Liouville integral to both sides of the FLDE then use a successive substitution (SS) to solve the resulting integral equation starting by the initial data. The simplicity and accuracy of this iterative scheme appears in the numerical and graphical results. This chapter is outlined in five sections. In Sect. 2, some basics and needed properties related to fractional calculus and generalised fractional power series are given. Section 3 consists out of three subsections. Each subsection is a brief description of one of the proposed techniques with the main definitions, theorems and algorithms to construct analytic and approximate solutions of the FLDE. In Sect. 4, two examples are carried out to illustrate the reliability and the simplicity of the proposed methods and to display how the fractional operator affects the behaviour of the solution of the population growth curves. Finally, a brief conclusion is presented in Sect. 5.

2 Preliminaries of Fractional Calculus This section is devoted to introducing some necessary definitions and mathematical preliminaries of fractional calculus, especially those related to Caputo operator, and basics of the generalised fractional power series. For more details, the reader can

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refer to [8, 11, 35, 42] for more details about fractional derivatives. Throughout this chapter, we deal with the following spaces:    • Lp [a, b] = ω : [a, b] → R : ab |ω(x)|p dx < ∞, , 1 ≤ p < ∞; • C[a, b] = {ω : [a, b] → R : ω is continuous on [a, b]}; • AC[a, b] = {ω : [a, b] → R : ω is absolutely continuous on [a, b]}; • AC n [a, b] = {ω : [a, b] → R : ω is n time absolutely continuous on[a, b]}. For the last two spaces, a function ω : [a, b] → R is called absolutely continuous, if ∀ > 0, there is δ > 0 such of disjoint intervals {[xi , yi ], i =  that for any finite set 1, 2, . . . , k} in [a, b] with ki=1 |yi − xi | < δ then ki=1 |ω(yi ) − ω(xi )| < . Among many definitions for the integrals and derivatives of fractional order, the most famous of them are the definitions of Riemann-Liouville fractional integral and Caputo fractional derivative which have some privacy. In this section, we give these definitions and some of their properties. Definition 1 The Riemann-Liouville fractional integral of order α > 0 over the interval [a, b] for a function ω ∈ L1 [a, b] is defined by α

Ja+ ω (t) =

1 Γ (α)



t a

ω(z) d ξ, t > a. (t − ξ )1−α

α For α = 0, Ja+ is the identity operator.

Definition 2 The Riemann-Liouville fractional derivative of order α for a function ω ∈ AC n [a, b], n ∈ N, t > a, n − 1 < α ≤ n, is defined by α

Da+ ω (t) =



d dt

n

n−α

Ja+ ω (t) =

 n  t d g(ξ) 1 d ξ. α−n+1 (n − α) dt a (t − ξ)

α

In particular, if 0 < α < 1, then Da+ ω (t) =

1 d (1−α) dt

t

ω(t) a (t−ξ)α d ξ,

t > a.

Definition 3 The Caputo fractional derivative of order α > 0 for a function ω ∈ AC n [a, b], n ∈ N, is defined by

α ω)(t) = J n−α Dn ω (t) = (C Da+ a+

 t ω(n) (ξ) 1 d ξ, t > a, n − 1 < α ≤ n. (n − α) a (t − ξ)α−n+1

α Specifically, for 0 < α < 1, and ω(t) ∈ AC[a, b], we have (C Da+ ω)(t) =   t ω (ξ) 1 d ξ. (1−α) a (t−ξ)α

Some properties of the Caputo fractional derivative and Riemann-Liouville integral are given in the following theorem [11]: Theorem 1 Let ω ∈ AC n [a, b] and > 0. Then,

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n−1 ω(k) (a) (t − a)k . k=0 k!

In particular, if 0 < α ≤ 1 and f ∈ AC[a, b] or ω is continuous on [a, b], then αC α (Ja+ Da+ ω)(t) = ω(t) − ω(a).

The Caputo fractional derivative can be easily computed for any function ω(t) through the formula C

α Da+ ω(t) =

∞  n=0

ω(n) (ξ ) (t − ξ )nα , 0 ≤ ξ < t, α > 0. Γ (n + 1 − α)

The next theorem shows the relations between Caputo and Riemann-Liouville fractional derivatives. Theorem 2 Let α > 0 and n − 1 < α ≤ n for n ∈ N. Then (

C

α Da+ ω)(t)

   n−1 ω(k) (a) α k (t). = Da+ ω(x) − (x − a) k=0 k!

However, since the Caputo fractional derivative will be only used in this chapter α with a = t0 = 0, then the symbol Dα will be instead of C Da+ . Definition 4 A power series (PS) expansion at t = t0 of the following form ∞ 

am (t − t0 )mα = a0 + a1 (t − t0 )α + a1 (t − t0 )2α + · · · ,

m=0

for −1 < β ≤ n, n ∈ N and t ≤ t0 , is called the fractional power series (FPS) [48].  mα Theorem 3 There are only three possibilities for the FPS ∞ m=0 am (t − t0 ) . They are as follows: (1) The series converges only for t = t0 . That is, the radius of convergence equals to zero. (2) The series converges for all t ≥ t0 . That is, the radius of convergence equals infinity. (3) The series converges for t ∈ [t0 , t0 + R), for some positive real number R and diverges for > t0 + R. Here, R is the radius of convergence for the FPS. Theorem 4 Suppose that ω(t) has FPS representation at t = t0 of the form ω(t) =

∞  m=0

cm (t − t0 )mα .

(6)

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If ω(t) ∈ C[t0 , t0 + R), and Dmα ω(t) ∈ C(t 0 , t0 + R), for m = 0, 1, 2, . . ., then Dtmα ω(t0 ) , where Dmα = Dα · Dα · · · Dα coefficients cm will be in the form cm = Γ (mβ+1) (m-times) [49].

3 Description of FRPS, RKHS and SS Approaches In this section, we give a brief description for the three proposed techniques that we will apply to solve fractional LDE. For each method, we present some necessary definitions and prove some convergence results. We summarise each proposed procedure by an algorithm.

3.1 The Reproducing Kernel Hilbert Space Method Definition 5 Let S be any nonempty abstract set. Then a function K : S × S → C is called a reproducing kernel of the Hilbert space H if and only if (a) ∀t ∈ S, K(·,) ∈ H, (b) ∀t ∈ S, ∀ω ∈ H, (ω(·), K(·, t)) = ω(t). The condition in (b) is called “the reproducing property” because it indicates that the value of the function ϕ at the point t is reproduced by the inner product of ω with K(·, t) [34]. The function K is called the reproducing kernel function of H. This function possesses some important properties such as being unique, conjugate symmetric and positive-definite. A Hilbert space which possesses a reproducing kernel is called a reproducing kernel Hilbert space (RKHS). Definition 6 The space of functions W21 [a, b] is defined as   W21 [a, b] = ω : [a, b] → R : ω ∈ AC[a, b], ω ∈ L2 [a, b] . The inner product and norm are given, respectively, by ω1 , ω2 W21 =  b

  ω1 (t), ω1 (t) W21 , ω1 , ω2 ∈ ω1 W21 = a ω1 (t)ω2 (t) + ω1 (t)ω2 (t) dt and W21 [a, b] [33]. Theorem 5 The space W21 [a, b] is a complete RKHS with the reproducing kernel function Tt (s) such that Tt (s) =

1 [cosh(t + s − b − a) + cosh(|t − s| − b + a)]. 2sinh(b − a)

Definition 7 The space of real functions W22 [a, b] is defined as follows:

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  W22 [a, b] = ω : ω, ω ∈ AC[a, b], ω ∈ L2 [a, b], ω(a) = 0 . The inner product and the norm are, respectively, given by ω1 , ω2 W22 =  ω1 (t)ω2 (t)dt and ω1 W22 = ω1 (t), ω1 (t) W22 , ω1 , ω2 ∈ W22 [a, b] [34].

b ω1 (a)ω2 (a)+ a

Theorem 6 The space W22 [a, b] is a RKHS and its reproducing kernel function Kt (s) has the form Kt (s) =

− a) 2a2 − s2 + 3t(2 + s) − a(6 + 3t + s) , s ≤ t, a) 2a2 − t 2 + 3s(2 + t) − a(6 + 3s + t) , s > t.

1

6 (s 1 − 6 (t

Now, let us consider the FLDE in (3) with the initial condition (4). To solve this equation using the RKHSM, we first homogenise the initial condition using the substitution P(t) = ω(t) − ω0 to get Dα P(t) + Dα ω0 = δ(P(t) + ω0 )(1 − P(t) − ω0 ). But Dα ω0 = 0, so Eqs. (3) and (4) becomes Dα P(t) = δ(P(t) + ω0 )(1 − P(t) − ω0 ), P(0) = 0.

(7)

Now, by defining the differential operator L : W22 [a, b] → W21 [a, b] such that LP(t) = Dα P(t), the Eq. (7) can be rewritten as LP(t) = δ(P(t) + ω0 )(1 − P(t) − ω0 ), t > 0. Theorem 7 The operator L is a bounded linear from W22 [a, b]to W21 [a, b] such that LP(t) = Dα P(t). Proof The linearity of the operator L results from the linearity of the Caputo derivative, so it is clear. For boundedness, we need to find M > 0 such that LPW21 ≤ M PW22 ∀P ∈ W22 [a, b]. To do this, we have  LP2W 1 2

= LP, LP W21 =

b



 2  [(LP)(t)]2 + (LP) (t) dt.

a

By the reproducing property of Kt (s), we can write P(t) = P(.), Kt (.) W22 and (Dα P)(t) = P(.), (Dα P)(.) W22 . Using Schwarz Inequality and the fact that and uniformly Dα Kt (s) is continuous   bounded in s and t, we get |(LP)(t)| =   α α 2 |(D P)(t)| =  P(.), (D Kt )(.) W2  ≤ PW22 (Dα Kt )W22 = M1 PW22 and           (LP) (t) =  d (Dα P)(t) =  P(.), d (Dα Kt )(.) 2  ≤ PW 2  d (Dα Kt ) 2 = dt dt dt W W 2 2

2

M2 PW22 , where M1 , M2 ∈ R+ .

Therefore, LP2W 1 ≤ (M1 )2 + (M2 )2 (b − a)P2W 2 = LPW21 ≤ M PW22 , 2

2

Solutions of Fractional Verhulst Model …

where Mjr =

241



M12 + M22 (b − a). Hence, L is bounded.

Now, we construct an orthogonal function system of the space W22 [a, b] by considering the countable dense set {ti }∞ i=1 of the interval [a, b], and let ϕi (t) = Tti (t) and ψi (t) = L∗ ϕi (t), where L∗ is the adjoint operator of L. In terms of the properties of the reproducing kernel Tt (.), we have P(t), ψi (t) W22 = P(t), L∗ ϕi (t) W22 = LP(t), ϕi (t) W21 = LP(ti ), i = 1, 2, . . . . Using the Gram-Schmidt orthogonalisa ∞ tion process on {ψi (t)}∞ i=1 , we can form the orthonormal function system ψi (t) i=1  of W22 [a, b] by defining ψi (t) = il=1 βil ψl (t), i = 1, 2, 3, . . ., where βil are the orthogonalisation coefficients which are given by: β11 =

1 , ψ1 W22

1 βii =  2 , and   ψi 2W 2 − i−1 (t), ψ (t) ψ i p p=1 W2 2

i−1 

2



− p=1 ψi (t), ψp (t) W 2 βpl 2 βil =  2 , for i > l. i−1  2 ψi W − p=1 ψi (t), ψp (t) W 2 2

Theorem 8 If {ti }∞ i=1 is dense on [a, b] and assuming that the solution of Eq. (7) is unique, then it has the form: P(t) = δ

∞  i 

βil (P(tl ) + ω0 )(1 − P(tl ) − ω0 )ψi (t).

i=1 l=1

The n-term approximate solution P n (t) of P(t) is given by the finite sum: P (t) = δ n

n  i 

βil (P(tl ) + ω0 )(1 − P(tl ) − ω0 )ψi (t).

i=1 l=1

 ∞ Proof According to the orthonormal basis ψi (t) i=1 of W22 [a, b], we have P(t) =

∞  

 P(t), ψi (t) W 2 ψi (t) 2

i=1

=

∞  i 

  βil P(t), L∗ ϕl (t) W 2 ψi (t) 2

i=1 l=1

=

∞  i  i=1 l=1

βil δ(P(t) + ω0 )(1 − P(t) − ω0 ), ϕl (t) W22 ψi (t)

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=δ

∞  i 

βil (P(tl ) + ω0 )(1 − P(tl ) − ω0 )ψi (t).

i=1 l=1

Theorem 9 For any function ω(t) in W22 [a, b], the approximate series ωn (t) and are uniformly convergent to ω(t) and ω (t), respectively.

d ωn dt

According the previous discussion, the approximate solution of (3) and (4) is ωn (t) = δ

n  i 

βil (P(tl ) + ω0 )(1 − P(tl ) − ω0 )ψi (t) + ω0 .

i=1 l=1

It is worth to mention that we can define another operator in order to solve the α to the non-linear FLDE by applying the Riemann-Liouville fractional integral J0+ α α α two sides of Eq. (7). So J0+ [D P](t) = J0+ [δ(P(t) + ω0 )(1 − P(t) − ω0 )] which are equivalent to P(t) =

δ t (P(s) + ω0 )(1 − P(s) − ω0 ) ∫ ds, t > 0. (α) 0 (t − s)1−β

(8)

Define the operator I : W22 [a, b] → W21 [a, b] by IP(t) = P(t). Obviously, I is a bounded linear operator. Thus, (7) can be rewritten as IP(t) = δ(P(t) + ω0 )(1 − P(t) − ω0 ). Applying the RKHS method with the operator I, we can obtain approximate solution of (3) and (4) of the form δ  ∗ P (t) = βil (P(tl ) + ω0 )(1 − P(tl ) − ω0 )ψi (t), (α) i=1 n

i

n

l=1

which converges to the analytic solution ∞

P n (t) =

δ  ∗ βil (P(tl ) + ω0 )(1 − P(tl ) − ω0 )ψi (t). (α) i=1 i

l=1

Algorithm 1 To approximate the solution of the LFDE (3) and (4), we do the following steps: Step 1: Put P(t) = ω(t) − ω0 to homogenise the initial condition. Step 2: Define the reproducing function over [a, b] as

If s ≤ t, set Kt (s) = 16 (s − a) 2a2 − s2 + 3t(2 + s) − a(6 + 3t + s) ;

else set Kt (s) = 16 (t − a) 2a2 − t 2 + 3s(2 + t) − a(6 + 3s + t) . Step 3: To obtain orthogonal function system ψij (t) for i = 1, 2, . . . , n, do the following: Set ti = a +

i−1 (b − a) so that ψi (t) = Dα [Kt (s)]s=ti ; n−1

Solutions of Fractional Verhulst Model …

243

Step 4: For i = 1, . . . , n and l = 1, 2, . . . , i − 1, set: β11 =

1 , ψ1 W22

1 βii =  2 , and i−1  2 ψi W 2 − p=1 ψi (t), ψp (t) W 2 2

βil = 

i−1 

2



ψi (t), ψp (t) W 2 βpl 2 2 , for i > l.   ψi 2W − i−1 ψ ψ p=1 i (t), p (t) W 2 −

p=1

2

 Step 5: For i = 1, . . . , n, ψi (t) = il=1 βil ψi (t) to get the orthonormal function system ψi (t).   Step 6: Set ωn (t) = δ ni=1 il=1 βil (P(tl ) + ω0 )(1 − P(tl ) − ω0 )ψi (t) + ω0 to obtain the approximate RKHS solution for the FLDE.

3.2 The Fractional Residual Power Series Method In this subsection, we describe the main steps needed to implement the FRPS algorithm in order to solve the continuous logistic equation in the fractional sense by expanding FPS and utilising repeated fractional differentiation of the truncated residual functions. This analysis aims to extend the application of fractional Taylor series framework to get an accurate analytic series solution of fractional system (3) and (4). To perform this, let the fractional logistic equation (3) and (4) has the solution form about t = 0: ω(t) =

∞ 

an

n=0

t nα . Γ (nα + 1)

(9)

Thus, if we use the initial data given by (4), D0α ω(0) = ω0 , as initial truncated series of ω(t), so the FPS solution of Eq. (3) can be written by ω(t) = ω0 +

∞ 

an

n=1

t nα . Γ (nα + 1)

(10)

Therefore, the j-th truncated series solution of ω(t) is given by ωj (t) = ω0 +

j  n=1

an

t nα . Γ (nα + 1)

(11)

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S. Hasan et al. j

According the FRPS approach, we define the j-th residual function, Resω (t), for the proposed logistic model as follows

Resωj (t) = D0α ωj (t) − δωj (t) 1 − ωj (t) , j = 1, 2, 3, . . . ,

(12)

whereas the residual function, Resω (t), can be defined by Resω (t) = lim Resωj (t) = D0α ω(t) − δω(t)(1 − ω(t)). j→∞

In this point, we noted that Resω (t) = 0 for all t ≥ 0, which leads to D0kα Resω (0) = = 0, for all k = 1, 2, . . . , j. Consequently, the following fractional relations assist us to determine the unknown coefficients, an , n = 1, 2, . . . , j, of Eq. (10)

j D0kα Resω (0)

(j−1)α

D0

ω(0) = 0, j = 1, 2, 3, . . . .

(13)

To show the iteration concept of the FRPS technique to find out a1 , follow the steps: α

t • Let ω1 (t) = ω0 + a1 Γ (α+1) ; • Substitute in Eq. (12) at j = 1 to get that

  δ(2ω0 − 1) α δa2 Resω1 (t) = δ(ω0 − 1)ω0 + a1 1 + t + 2 1 t 2α . Γ (α + 1) Γ (α + 1)

(14)

Set Resω1 (0) = 0, which yields

a1 = δω0 (1 − ω0 ). Thus, we get the first FRPS approximation for Eqs. (3) and (4) as   δ(1 − ω0 ) α t . ω1 (t) = p0 1 + Γ (α + 1) In the same way, to find out a2 , follow the steps: • Put the second truncated series   δ(1 − ω0 ) α t 2α ω2 (t) = ω0 1 + t + a2 ; Γ (α + 1) Γ (2α + 1)

(15)

Solutions of Fractional Verhulst Model …

245

• Substitute it in Resω2 (t) such as

Resω2 (t) = D0α ω2 (t) − δω2 (t)(1 − ω2 (t)) = δω0 (1 − ω0 )    δ 2 ω02 (1 − ω0 )2 2α 2δ δ(2ω0 − 1) α 3α + t + a2 t 1+ 1+ t Γ (α + 1) Γ (α + 1)Γ (2α + 1) Γ 2 (α + 1)   1 δ δ(2ω0 − 1) 2α + a22 2 tα + t t 4α . + a2 (16) Γ (α + 1) Γ (2α + 1) Γ (2α + 1) • Apply the operator D0α on both sides of Eq. 16 to get

λδ 3 Γ (2α + 1)ω02 (1 − ω0 )2 α D0α Resω2 (t) = t + δ 2 ω0 (1 − ω0 ) Γ 3 (α + 1)   2Γ (3α + 1) 2α t (2ω0 − 1) + a2 Γ (α + 1)Γ 2 (2α + 1)   δ(2ω0 − 1) α δΓ (4α + 1)a2 t + 2 t 3α . + a2 1 + Γ (α + 1) Γ (2α + 1)Γ (3α + 1) • Use the fact D0α Resω2 0 (0) = 0 in the above equation, to obtain

a2 = −δ 2 ω0 (1 − ω0 )(2ω0 − 1). • Rewrite the second FRPS approximation as   (ω0 − 1) α 2 (ω0 − 1)(2ω0 − 1) 2α . t +δ t ω2 (t) = ω0 1 − δ Γ (α + 1) Γ (2α + 1) Similarly, we have  t nα • Let ω2 (t) = ω0 + 3n=1 an Γ (nα+1) ; • Substitute it into the residual function Resω3 (t) such that

Resω3 (t) = D0α ω3 (t) − δω3 (t)(1 − ω3 (t)). • Apply D02α to Resω3(t). • Solve D02α Resω3 (t)t=0 = 0 for a3 to get

(17)

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S. Hasan et al.

a3 = δ 3



ω0 (1 − ω0 ) Γ 2 (α + 1) − ω0 (1 − ω0 ) 4Γ 2 (α + 1) + Γ (2α + 1) . Γ 2 (α + 1)

• The third FRPS approximation is

(ω0 − 1)(2ω0 − 1) 2α (ω0 − 1) α t + δ2 t Γ (α + 1) Γ (2α + 1)

(ω0 − 1) Γ 2 (α + 1) + ω0 (ω0 − 1) 4Γ 2 (α + 1) + Γ (2α + 1) 3α t . − δ3 Γ 2 (α + 1)Γ (3α + 1)

ω3 (t) = ω0 (1 − δ

Further, when the same routine is repeated as above up to the arbitrary order k, the coefficients an , n = 4, 5, 6, . . . , k, can be obtained. Anyhow, the next algorithm summarises the procedure in the previous discussion to find the unknown coefficients of the fractional power series that represents the solution of the FLDE. Algorithm 2 To determine the required coefficients of ωj (t) in (11) which represents the analytic solution of (3) and (4), do the following steps: Step 1: Consider the initial condition ω0 = ω(0), which is the zeroth FPS approximate solution of ω(t). j j Step 2: Substitute ωj (t) = ω0 + n=1 an t nα into the jth -residual function Resω (t). j (j−1)α Resω (t) for j = 1, 2, . . . , k. Step 3: Compute Dt j (j−1)α Resω (0) = 0 for aj . Step 4: Solve the resulting fractional equation Dt Step 5: Substitute the obtained coefficients, for j = 1, 2, . . . , k, back into Eq. (7) which yields the required analytic solution. jα

Lemma 1 Suppose that ω(t) ∈ C[0, R), R > 0, D0 ω(t) ∈ C(0, R), and 0 < α ≤ 1. Then for any j ∈ N, we have jα



D ω(0) jα jα jα (j+1)α (j+1)α J0 D0 ω (t) − J0 t . D0 ω (t) = 0 (jα + 1)

Theorem 10 Let ω(t) has the FPS in (7) with radius of convergence R > 0, jα and suppose that ω(t) ∈ C[0, R), D0 ω(t) ∈ C(0, R)for j = 0, 1, 2, . . . , N + 1. N D0jα ω(0) jα and RN (ζ ) = Then, ω(t) = ωN (t) + RN (ζ ), where ωN (t) = j=0 (jα+1) t D0(N +1)α ω(ζ ) (N +1)α t , ((N +1)α+1)

for some ζ ∈ (0, t).

In view of the previous theorem, we can consider the formula of ωN (t) as an approximation of ω(t), and RN (ζ ) as the truncation (remainder) error that results from approximating ω(t) by ωN (t). Also, the upper bound of the error can be predicted by

Solutions of Fractional Verhulst Model …

247

    Mt (N +1)α  , |RN (ζ )| ≤ Supt∈[0,R] ((N + 1)α + 1)      provided that D0(N +1)α ω(ζ ) < M on [0, R).

3.3 The Successive Substitutions Technique In this subsection, we provide a simple iterative scheme to solve system (3) and (4). α to both sides of Eq. (3), To do this, by applying the Riemann-Liouville operator J0+ the following Volterra integral equation can be obtained δ ω(t) = ω0 + (α)

t

ω(ξ )(1 − ω(ξ ))(t − ξ )α−1 d ξ.

0

In order to approximate the solution of Volterra integral equation, and consequently the solution of the FLDE, we may apply a successive substitution (SS) technique as follows. Let ω0 (t) = ω0 = ω(0), δ ωn+1 (t) = ω0 + (α)



t

ωn (ξ )(1 − ωn (ξ ))(t − ξ )α−1 d ξ, n = 0, 1, 2, . . .

0

Hence, the first approximation is ω1 (t) = ω0 1 − approximation is

t α δ(−1+ω0 ) (α+1)

, and the second

  ⎞ αδ 4α t 3α δ 3  21 + α (η − 1)η t 2α δ 2 (2η − 1) t ⎠. − √ + ω2 (t) = ω0 (ω0 − 1)⎝1 − [1 + α] [1 + 2α] π [1 + α][1 + 3α] ⎛

Continuing this process, we may obtain the exact solution as ω(t) = limn→∞ ωn (t). Algorithm 3 To determine the nth SS approximate solution ωn (t) for Eq. (3) and (4), do the following two steps: Step 1: Consider the initial condition ω0 = ω(0) as the zeroth SS approximate solution of ω(t). Step 2: For k = 1, 2, . . . , n, substitute

δ ωk (t) = ω0 + (α)

t 0

ωk−1 (ξ )(1 − ωk−1 (ξ ))(t − ξ )α−1 d ξ.

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S. Hasan et al.

4 Numerical Applications In this section, we compute approximate solution for the FLDE using three numerical approaches: the RKHS, the FRPS and the SS methods. Using these techniques, we study two examples of the FLDE. For each example, we apply the proposed methods for different values of the fractional derivative. To show the efficiency and accuracy of these methods, we compare the exact with the approximate solutions in the case of integer-order derivative and compute absolute and relative errors. Moreover, we compare the numerical solutions for different orders of the Caputo derivative in graphs and tables. We observe that all the three techniques are convenient methodologies for controlling the convergence of the solution and the results are found in good agreement. The graphical results show that the variety of choice of fractional orders leads to a variety in predicted curves for the population growth, which of course, leads to better results. All computations are performed using Mathematica 10. Example 1 Consider the following non-linear FLDE: D0α ω(t) =

1 ω(t)(1 − ω(t)), t ∈ [0, 1], 0 < α ≤ 1, 3

(18)

subject to the initial condition ω(0) =

1 . 3

(19) t/3

e The exact solution of the above IVP is ω(t) = 2+e t/3 . Following Algorithm 1, we have to homogenise the IC to get

D0α P(t) =

   1 2 1 P(t) + − P(t) , P(0) = 0, t ∈ [0, 1], 0 < α ≤ 1. 3 3 3

Taking n = 25, Table 1 shows the accuracy of this technique. While, the approximate solution of the FLDE for different values of α is given in Table 2. On the other hand, following the procedures outlined in this chapter for the FRPS method, we take six iterations and summarise some of the results in Tables 3 and 4. The fourth FPS for α = 1, α = 0.8, α = 0.5 are listed below: 1 + 0.0740741 t + 0.00411523 t 2 − 0.000457247 t 3 3 − 0.0000635066 t 4 ;

ω4,α=1 (t) =

1 + 0.0795312 t 0.8 + 0.00575707 t 1.6 − 0.000704323 t 2.4 3 − 0.000147395 t 3.2 ;

ω4,α=0.8 (t) =

Solutions of Fractional Verhulst Model …

249

Table 1 Absolute and relative errors for Example 1 for α = 1 using RKHS method t

RKHS solution

Absolute error

0.0

0.333333333

0.0

0.1 0.2

0.340781423 0.348308985

Relative error 0.0

6.55734649 ×

10−9

1.92420887 × 10−8

1.31846654 ×

10−8

3.78533586 × 10−8

10−8

5.52014396 × 10−8

0.3

0.355913052

1.96469139 ×

0.4

0.363590507

2.59394049 × 10−8

7.13423552 × 10−8

0.371338094

3.20587129 ×

10−8

8.63329421 × 10−8

10−8

1.00230654 × 10−7

0.5 0.6

0.379152415

3.80026985 ×

0.7

0.387029942

4.37705229 × 10−8

1.13093363 × 10−7

0.394967015

4.93626510 ×

10−8

1.24979157 × 10−7

10−8

1.35946132 × 10−7 1.46052192 × 10−7

0.8 0.9

0.402959856

5.47808413 ×

1.0

0.411004569

6.00281270 × 10−8

Table 2 RKHS solutions of Example 1 for different values of fractional derivative t

α = 0.9

α = 0.8

α = 0.7

α = 0.6

α = 0.5

α = 0.4

0.0

0.333333

0.333333

0.333333

0.333333

0.333333

0.333333

0.1

0.342354

0.344504

0.347041

0.349995

0.353382

0.357204

0.2

0.351427

0.355279

0.359757

0.364896

0.370713

0.377189

0.3

0.359926

0.364564

0.369790

0.375611

0.382003

0.388904

0.4

0.367993

0.372816

0.378034

0.383607

0.389459

0.395473

0.5

0.375912

0.380739

0.385766

0.390909

0.396053

0.401041

0.6

0.383753

0.388472

0.393218

0.397879

0.402311

0.406341

0.7

0.391531

0.396018

0.400374

0.404473

0.408164

0.411274

0.8

0.399254

0.403388

0.407251

0.410711

0.413620

0.415818

0.9

0.406929

0.410611

0.413893

0.416646

0.418731

0.420003

1.0

0.414579

0.417755

0.420409

0.422419

0.423656

0.423993

1 + 0.0835836 t 0.5 + 0.00823045 t − 0.00106387 t 1.5 3 − 0.000383401 t 2 .

ω4,α=0.5 (t) =

For the last technique, the FLDE can be solved using SS method with ω0 = 13 . The first two SS iterations are ⎞ ⎛ √ 2t α 8t 3α (2α) 1 1 4−α π t 2α ⎠ 54 ⎝

. ω1 (t) = + 9+ + and ω2 (t) = − 3 27(α + 1) 3 6561α 2 (α) ((α + 1))2 (3α)  α + 21

The fourth iteration for α = 1, α = 0.8, α = 0.5 are listed below:

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S. Hasan et al.

Table 3 Absolute and relative errors for Example 1 for α = 1 by FRPS method t

FRPS solution

0.0

0.333333333

0.1 0.2

0.340781429 0.348308998

Absolute error

Relative error

0.0

0.0

1.27675647 ×

10−15

3.746555323 × 10−15

1.48991929 ×

10−13

4.277579122 × 10−13

10−12

6.66873595 × 10−12

0.3

0.355913071

2.37349029 ×

0.4

0.363590533

1.64749880 × 10−12

4.531192797 × 10−11

0.371338125

7.22676918 ×

10−11

1.946142527 × 10−10

10−10

6.229493852 × 10−10

0.5 0.6

0.379152452

2.36192787 ×

0.7

0.387029984

6.27320917 × 10−10

0.8

0.394967063

1.620858696 × 10−9

1.424094787 ×

10−9

3.605603894 × 10−9

10−9

7.071530277 × 10−9 1.245501315 × 10−8

0.9

0.402959908

2.849543213 ×

1.0

0.411004623

5.119068058 × 10−9

Table 4 FRPS solutions of Example 1 for different values of the fractional derivative t

α = 0.9

α = 0.8

α = 0.7

α = 0.6

α = 0.5

α = 0.4

0.0

0.3333333

0.33333333

0.33333333

0.33333333

0.33333333

0.333333

0.1

0.3431061

0.34607989

0.34985592

0.35461228

0.36055037

0.367887

0.2

0.3516901

0.35570238

0.36042250

0.36592033

0.37224872

0.379427

0.3

0.3599338

0.36448506

0.36958364

0.37522532

0.38137402

0.387947

0.4

0.3679885

0.37278715

0.37795795

0.38344549

0.38915855

0.394960

0.5

0.3759200

0.38076197

0.38580186

0.39094948

0.39608036

0.401030

0.6

0.3837683

0.38849262

0.39325249

0.39793077

0.40238497

0.406441

0.7

0.3915414

0.39603018

0.40039364

0.40450624

0.40821844

0.411358

0.8

0.3992664

0.40340854

0.40728107

0.41075307

0.41367612

0.415887

0.9

0.4069478

0.41065145

0.41395408

0.41672549

0.41882433

0.420102

1.0

0.4145917

0.41777615

0.42044161

0.42246332

0.42371145

0.424056

2t t2 t3 5t 4 t5 41t 6 1 + + − − + + 3 27 243 2187 78732 531441 47829690 1013t 8 26t 9 61t 7 − + − 4519905705 162716605380 3295011258945 349t 11 584t 10 − + 16475056294725 4893091719533325 688t 13 38t 12 + − 266895911974545 281041395309195885 256t 15 64t 14 − . + 272393967761220627 36773185647764784645

ω4,α=1 (t) =

Solutions of Fractional Verhulst Model …

251

1 + 0.0795312 t 0.8 + 0.00575707 t 1.6 − 0.000704323 t 2.4 3 − 0.000147395 t 3.2 + 2.7352 × 10−6 t 4 + 3.57138 × 10−6 t 4.8

ω4,α=0.8 (t) =

− 3.92022 × 10−8 t 5.6 − 4.3661 × 10−8 t 6.4 − 3.6982 × 10−10 t 7.2 + 4.32763 × 10−10 t 8 + 2.36072 × 10−12 t 8.8 − 3.09023 × 10−12 t 9.6 + 5.81878 × 10−14 t 10.4 + 8.82091 × 10−15 t 11.2 −3.24929 × 10−16 t 12 . 1 + 0.0835836t 0.5 + 0.00823045t − 0.00106387t 1.5 3 − 0.000383401t 2 − 6.94706 × 10−6 t 2.5 + 0.0000194138t 3

ω4,α=0.5 (t) =

+ 4.15567 × 10−8 t 3.5 − 4.41534 × 10−7 t 4 − 1.40437 × 10−8 t 4.5 + 8.61155 × 10−9 t 5 + 2.03093 × 10−10 t 5.5 − 1.25157 × 10−10 t 6 + 2.47645 × 10−12 t 6.5 + 6.9536 × 10−13 t 7 − 3.33294 × 10−14 t 7.5 . The errors when applying this iterative method are shown in Table 5, while numerical values are given in Table 6. For these results, we take six successive substitutions. To notice if there is difference between results from these three procedures, we graphed the approximate solutions in the same plane in Fig. 1. Also, we graph the absolute errors that result from applying each of these numerical procedures in Fig. 2. Obviously, the solution behaviour indicates that an increase of the fractional parameter changes the nature of the solution. Also, from the present results, we can conclude that all three proposed algorithms are convenient to solve FFDEs and in good agreement. Table 5 Absolute and relative errors for Example 1 for α = 1 using SSM t

SS solution

Absolute error

Relative error

0.0

0.333333333

0.0

0.0

0.1

0.340781429

0.0

0.0

0.2

0.348308998

2.220446049 × 10−16

6.374931627 × 10−16

0.355913071

3.885780586 ×

10−15

1.091777993 × 10−14

10−14

6.778765923 × 10−14

0.3 0.4

0.363590533

2.464695115 ×

0.5

0.371338125

9.908740495 × 10−14

2.668387599 × 10−14

0.379152453

2.988165271 ×

10−13

7.881170876 × 10−13

10−13

1.898276915 × 10−12

0.6 0.7

0.387029985

7.346900865 ×

0.8

0.394967064

1.553090989 × 10−12

3.932203790 × 10−12

0.402959911

2.919331443 ×

10−12

7.244719294 × 10−12

4.992062319 ×

10−12

1.214600023 × 10−11

0.9 1.0

0.411004629

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S. Hasan et al.

Table 6 SS solutions of Example 1 for different values of the fractional derivative t

α = 0.9

α = 0.8

α = 0.7

α = 0.6

α = 0.5

α = 0.4

0.0

0.333333

0.3333333

0.3333333

0.3333333

0.3333333

0.3333333

0.1

0.343106

0.3460798

0.3498559

0.3546122

0.3605503

0.3678867

0.2

0.351690

0.3557023

0.3604225

0.3659203

0.3722487

0.3794268

0.3

0.359933

0.3644850

0.3695836

0.3752253

0.3813740

0.3879473

0.4

0.367988

0.3727871

0.3779579

0.3834455

0.3891586

0.3949600

0.5

0.375920

0.3807619

0.3858018

0.3909495

0.3960804

0.4010302

0.6

0.383763

0.3884926

0.3932525

0.3979308

0.4023852

0.4064412

0.7

0.391541

0.3960301

0.4003936

0.4045063

0.4082188

0.4113584

0.8

0.399266

0.4034085

0.4072811

0.4107533

0.4136767

0.4158884

0.9

0.406947

0.4106515

0.4139542

0.4167258

0.4188252

0.4201042

1.0

0.414591

0.4177762

0.4204418

0.4224639

0.4237127

0.4240585

Fig. 1 A comparison between the RKHS, FRPS and SS solutions of Example 1 for different orders fractional derivatives: … RKHS ωn (t), FRPS ωn (t), … SS ωn (t)

Example 2 Consider the following non-linear FLDE: D0α ω(t) =

1 ω(t)(1 − ω(t)), t ∈ [0, 1], 0 < α ≤ 1, 2

(20)

Solutions of Fractional Verhulst Model …

253

Fig. 2 Absolute errors: (a) RKHS, (b) FRPS and (c) SS methods for Example 1

subject to the initial condition ω(0) =

1 . 5

(21) t/2

e The exact solution of the above IVP is ω(t) = 4+e t/2 . Following the Algorithm 1, we have to homogenise the IC to get

D0α P(t) =

   1 4 1 P(t) + − P(t) , P(0) = 0, t ∈ [0, 1], 0 < α ≤ 1. 2 5 5

Taking n = 25, Table 7 shows the accuracy of this technique. For different values of α, the approximate solution of the FLDE is given in Table 8. On the other hand, following the procedures outlined in this chapter for the FRPS method, we take six iterations and summarise some of the results in Tables 9 and 10. For the last technique, the FLDE can be solved using SS method with ω0 = 15 . The first three SS iterations are ω1 (t) =

2t α 1 + , 5 25(α + 1)

! √ tα 50 154−α π t α 4t 2α [2α] 1

− + ω2 (t) = + , 5 625 (α + 1) (α + 1) α + 21 ((α + 1))2 (3α + 1)

⎛ 523+2α t 2α  α + 21 −23625t 3α (3α) 3000 1 tα − ω3 (t) = + + 875⎝ √ 2 5 32812500(α) α 2 (2α)(4α) α π α (3α)

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Table 7 Absolute and relative errors for Example 2 for α = 1 using RKHS method t

RKHS solution

Absolute Error

Relative Error

0.1

0.208120076

3.379174099 × 10−8

1.623665343 × 10−7

0.2

0.216480619

6.994193039 × 10−8

3.230862332 × 10−7

0.225081557

1.065140960 ×

10−7

4.732242243 × 10−7

10−7

6.128886908 × 10−7

0.3 0.4

0.233922198

1.433683575 ×

0.5

0.243001189

1.803603639 × 10−7

7.422195346 × 10−7

0.252316499

2.173427363 ×

10−7

8.613885731 × 10−7

10−7

9.705994472 × 10−7

0.6 0.7

0.261865377

2.541666377 ×

0.8

0.271644341

2.906834533 × 10−7

1.070087236 × 10−6

0.281649145

3.267465498 ×

10−7

1.160117744 × 10−6

3.622130897 ×

10−7

1.240986467 × 10−6

0.9 1.0

0.291874770

Table 8 RKHS solutions of Example 2 for different values of the fractional derivative t

α = 0.95

α = 0.85

α = 0.75

α = 0.65

α = 0.55

0.0

0.2

0.2

0.2

0.2

0.2

0.1

0.2085909388

0.2106501316

0.2130776939

0.2158996236

0.2191301069

0.2

0.2176968227

0.2215403565

0.2260325928

0.231221992

0.2371380454

0.3

0.2270357312

0.2320298817

0.2377706961

0.2443176899

0.2517120038

0.4

0.236430584

0.2420615120

0.2483834360

0.2554405964

0.2632591967

0.5

0.2457805209

0.2517473876

0.2582547508

0.2652963638

0.2728401322

0.6

0.2551847215

0.2612933947

0.2677531712

0.2744944401

0.2814094655

0.7

0.2647676723

0.2708877764

0.2771755006

0.2835098158

0.2897209014

0.8

0.2745550464

0.2805544582

0.2865470405

0.2923758538

0.2978301946

0.9

0.2845305358

0.2902745497

0.2958400931

0.3010472160

0.3056651410

1.0

0.2946087218

0.3000222144

0.3050864641

0.3096053875

0.3133396499

⎛

⎞

2 41+2α t 2α (6α) α + 21 94α t 3α  α + 21 1 ⎜ 4α ⎠+ − 4t ⎝− √ π (7α) α 2 π (4α) α 4 (α)(3α)2

⎞⎞  α α(5α) 3542+α α(4α) α + 21 75t α 175t ⎠⎠ + 3150t α (α) + + (3α) ⎝ √ (6α) (3α + 1) π (5α)

⎞⎞ √ 250 π (5α + 1) − 316α t 3α  2α + 21 ⎠⎠. + √ π (5α + 1)(2α + 1) ⎛

The errors when applying this iterative method are shown in Table 11, while a sample of the numerical results are given in Table 12. For these results, we take six successive substitutions. To notice if there is difference between results from these

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Table 9 Absolute and relative errors for Example 2 for α = 1 using FRPS method t

FRPS Solution

Absolute error

0.0

0.2

0.0

0.1

0.208120110

0.2

0.216480689

Relative error 0.0

8.00748356 ×

10−14

3.847529 × 10−13

1.03367869 ×

10−11

4.774923 × 10−11

10−10

7.906202 × 10−10

0.3

0.225081664

1.77954123 ×

0.4

0.233922340

1.34282263 × 10−9

5.7404633 × 10−9

0.243001363

6.44730971 ×

10−9

2.6531989 × 10−8

10−8

9.2158679 × 10−8

0.5 0.6

0.252316693

2.32531752 ×

0.7

0.261865563

6.88311416 × 10−8

2.6284908 × 10−7

0.271644455

1.76294589 ×

10−7

6.4898977 × 10−7

10−7

1.4352858 × 10−6 2.9101733 × 10−6

0.8 0.9

0.281649067

4.04247487 ×

1.0

0.291874283

8.49407245 × 10−7

Table 10 FRPS solutions of Example 2 for different values of the fractional derivative t

α = 0.95

α = 0.85

α = 0.75

α = 0.65

0.0

0.2

0.2

0.2

0.2

α = 0.55 0.2

0.1

0.209326063

0.2122614428

0.2160541797

0.2209413786

0.2272259965

0.2

0.218315534

0.2225536329

0.2276652747

0.2338068760

0.2411521445

0.3

0.227349931

0.2324254862

0.2382910547

0.2450298545

0.2527108790

0.4

0.236498213

0.2421252989

0.2484162179

0.2553899593

0.2630302820

0.5

0.245789092

0.2517561978

0.2582379505

0.2651976372

0.2725500299

0.6

0.255236644

0.2613715314

0.2678584462

0.2746131007

0.2814954330

0.7

0.264847780

0.2710017612

0.2773374643

0.2837308534

0.2899996493

0.8

0.274625211

0.2806649490

0.2867124049

0.2926113618

0.2981486946

0.9

0.284568847

0.2903716588

0.2960075485

0.3012954313

0.3060016177

1.0

0.294676544

0.3001275373

0.3052388626

0.3098116011

0.3136007800

Table 11 Absolute and relative errors for Example 2 for α = 1 using SSM t

SS solution ωn (t)

Absolute error

Relative error 10−15

4.93444049 × 10−15

0.1

0.2081201100

1.026956298 ×

0.3

0.2250816644

1.944999717 × 10−12

8.64130679 × 10−12

0.5

0.2430013702

5.729930419 ×

10−11

2.35798276 × 10−10 2.35798276 × 10−10

0.7

0.2618656315

4.906398066 × 10−10

1.873631919 × 10−9

0.9

0.2816494695

2.276855093 ×

10−9

8.084002712 × 10−9

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Table 12 SS solutions of Example 2 for different values of the fractional derivative t

α = 0.95

α = 0.85

α = 0.75

α = 0.65

α = 0.55

0.1

0.2093260630

0.2122614428

0.2160541797

0.2209413780

0.2272259993

0.2

0.2183155346

0.2225536331

0.2276652760

0.2338068718

0.2411521873

0.3

0.2273499315

0.2324254885

0.2382910663

0.2450298550

0.2527110946 0.2527110946

0.4

0.2364982168

0.2421253119

0.2484162714

0.2553900182

0.2630309643

0.5

0.2457891050

0.2517562474

0.2582381266

0.2651979050

0.2725517020

0.6

0.2552366879

0.2613716801

0.2678589132

0.2746138898

0.2814989163

0.7

0.2648479033

0.2710021379

0.2773385312

0.2837327154

0.2900061343

0.8

0.2746255118

0.2806657922

0.2867145888

0.2926151761

0.2981598119

0.9

0.2845695109

0.2903733762

0.2960116590

0.3013025036

0.3060195091

three procedures, we graphed the approximate solutions in the same plane in Fig. 3, and the absolute error curves for each method are shown in Fig. 4.

Fig. 3 A comparison between the RKHS, FRPS and SS solutions of Example 2 for different orders fractional derivatives: … RKHS ωn (t), __FRPS ωn (t), … SS ωn (t).

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Fig. 4 Absolute errors: (a) RKHS, (b) FRPS, (c) SS methods for Example 2

5 Conclusion In this work, we applied three analytic and numeric schemes in order obtain approximate solutions for the non-linear FLDE, which are the RKHS, the FRPS and the SS methods. The fractional derivative was described in the Caputo sense. Two examples were given to show the efficiency of the proposed methods. By comparing our results with the exact solution for integer order derivative, we observe that the proposed methods yield accurate approximations. To see the effects of the fractional derivative on the logistic curve, we solved the same FLDE for different values of the fractional order using these three procedures. The results showed that the curves of FLDE approach the curve of LDE as the fractional order approaches the integer order. Obviously, the solution behaviour indicates that an increase of the fractional parameter changes the nature of the solution. Also, from the present results, we can conclude that all three proposed algorithms are convenient to solve FFDEs and in good agreement. Funding: This research was funded by Ajman University, UAE (Grant ID 2020COVID 19-08: GL: 5211529). Acknowledgements The authors gratefully acknowledge support from Ajman University, UAE.

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Is It Worthwhile Considering Orthogonality in Generalised Polynomial Chaos Expansions Applied to Solving Stochastic Models? Julia Calatayud, J.-C. Cortés, Marc Jornet, and Laura Villafuerte

Abstract Generalised polynomial chaos (gPC) expansions allow approximating random variables and stochastic processes in a mean square sense. These expansions together with a stochastic Galerkin projection technique permit approximating the solution process to stochastic systems with statistically independent input random parameters. The expansions are constructed in terms of orthogonal polynomials that may belong to the Askey scheme or may be made from a Gram–Schmidt orthonormalisation procedure. When the random input coefficients are statistically dependent, a variation of the gPC method was proposed in a recent contribution, in which the stochastic processes are developed directly in terms of the canonical polynomial basis. This new approach was presented from a computational standpoint. In this chapter, we prove that both methods (gPC and canonical polynomial expansions) yield exactly the same results when the random inputs are independent. We comment on the advantages and disadvantages of both techniques, and we make suggestions for computational applications.

1 Introduction and Motivation Uncertainty quantification for stochastic models requires the computation of statistics of the output stochastic process. Although it would be desirable to have the exact expression of the statistics, in general, this is not possible, and approximations must

J. Calatayud · J.-C. Cortés · M. Jornet Instituto de Matemática Multidisciplinar, Universitat Politècnica de València, Camino de Vera, s/n 46022 Valencia, Spain URL: https://www.imm.upv.es/ L. Villafuerte (B) Department of Mathematics, University of Texas at Austin, Austin, USA e-mail: [email protected] URL: https://www.utexas.edu/ © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 D. Zeidan et al. (eds.), Computational Mathematics and Applications, Forum for Interdisciplinary Mathematics, https://doi.org/10.1007/978-981-15-8498-5_12

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be sought. One thus looks for faithful representations of the response process. For example, mean square expansions allow approximating its expectation and variance. Generalised polynomial chaos (gPC) expansions represent second-order random variables and stochastic processes as infinite series of orthogonal polynomials having mean square convergence. In the celebrated paper [1], the authors used these expansions together with a stochastic Galerkin projection technique to solve continuous stochastic systems when the random input coefficients are statistically independent and can be related to random variables that have probability distributions that belong to the Askey scheme. The key idea is that the weighting functions associated with the orthogonal polynomials are identical to the probability density functions of the corresponding random variables. This theory has had a great impact in stochastic modeling, see for example, [2–8]. When some transmission random parameter has a probability distribution that is not classified in the Askey scheme, a Gram–Schmidt orthogonalisation procedure was proposed in [9, 10]. The numerical experiments therein showed that rapid convergence of the Galerkin projections usually holds. This convergence was formalised in [11–14]. For statistically dependent random input coefficients, a computational approach was suggested in [15] and recently applied to discrete epidemiological models in [16]. The output stochastic process is expressed directly in terms of the canonical polynomial bases evaluated at the input parameters. The numerical experiments showed the good capability of the proposed method to approximate both the expectation and the variance. This chapter aims at comparing the methodologies proposed in [1, 9] (representations with orthogonal polynomials) and [15] (representations in terms of canonical polynomial bases). We will prove that both methods give exactly the same approximations, so that [15] generalises the classical gPC approach. However, expansions based on orthogonal polynomials have much more desirable numerical properties. The structure of the present chapter is the following. In Sect. 2, we expose the preliminaries on gPC and canonical polynomial expansions. Section 3 is devoted to compare both techniques, both from a theoretical and a numerical viewpoint. In Sect. 4, numerical experiments bring to light the theoretical discussion of the chapter. Finally, Sect. 5 summarises the main conclusions and details useful suggestions.

2 Preliminaries on gPC and Canonical Polynomial Expansions In this chapter, we deal with continuous stochastic systems defined by the following random differential equation problem: 

x  (t) = F(t, x(t), ζ ), x(t0 ) = x0 .

(1)

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The term ζ is a finite random vector and x0 is a random variable. We denote ξ = (ζ, x0 ) = (ξ1 , . . . , ξs ) ∈ Rs the vector of input random coefficients. The expression F is a deterministic real function F : [t0 , ∞) × R × Rs−1 → R. The term x(t) is the solution stochastic process, considered in a sample path or mean square sense (see [17, 18] for the treatment of solutions to random differential equations). All these random quantities are assumed to be defined in an underlying complete probability space (, F , P), where  is the sample space formed by outcomes ω ∈ , F ⊆ 2 is the σ -algebra of events and P is the probability measure. In our study of (1), we assume that the random vector ξ is absolutely continuous with probability density function f ξ (ξ ). We also assume that each ξi , 1 ≤ i ≤ s, has absolute moments of any order. This assumption holds for the common probability distributions, say Normal, Gamma, Beta, Uniform, Triangular, etc. The goal is to capture the uncertainty of x(t) (uncertainty quantification), by approximating its expectation and variance at each time instant t, E[x(t)] and V[x(t)]. It is assumed that x(t), which is a deterministic transformation of ξ , belongs to the space L2 (), which has the structure of a Hilbert space with the inner product X, Y = E[X Y ],

X, Y ∈ L2 ().

We show two techniques that allow approximating x(t) in the mean square sense. The first one is based on gPC expansions, in which we expand x(t) in terms of orthogonal polynomials constructed from the input random coefficients. The second method is based on expanding x(t) directly in terms of canonical polynomial bases evaluated at the input random parameters. (i) gPC approach [1, 9, 10, 19]: Assume that ξ1 , . . . , ξs are independent random variables, so that s  f ξ (ξ ) = f ξi (ξi ). i=1 p

p

Consider the canonical bases Ci = {1, ξi , . . . , ξi }, for p ≥ 1, 1 ≤ i ≤ s. By using a Gram–Schmidt orthogonalisation procedure, we obtain an orthogonal basis in L2 () p i = {φ0i (ξi ), . . . , φ ip (ξi )}. This approach is the foundation of the so-called adaptive gPC, see [9, 10]. Notice that, if ξi belongs to the Askey scheme, then one can directly use its associated family of orthogonal polynomials (for example, if ξi is Gaussian distributed, p then i is the family of Hermite polynomials; if ξi is uniformly distributed, then p i is the family of Legendre polynomials; etc.) [1, 19]. By performing a simple tensor product, we obtain an orthogonal basis in L2 ()  P = {φ0 (ξ ), . . . , φ P (ξ )},

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where φ0 = 1, P + 1 = ( p + s)!/( p!s!), and φ j (ξ ) = φi11 (ξ1 ) · · · φiss (ξs ), being i 1 , . . . , i s ≥ 0, i 1 + · · · + i s ≤ p and j ↔ (i 1 , . . . , i s ) in a bijective manner (for example, by means of the graded lexicographic order). Thus, we have constructed P in L2 (). Formally, when p → a family of orthogonal polynomials {φi (ξ )}i=0 ∞ 2 ∞, x(t) = i=0 xˆi (t)φi (ξ ) in L (), for each t, where xˆi (t) = E[x(t)φi (ξ )]/γi , being γi = E[φi (ξ )2 ] the normalisation constant. This is the gPC expansion of x(t). Some theoretical results in the literature justify the convergence of gPC expansions, under the assumption that the moment problem for each ξi , 1 ≤ i ≤ s, is uniquely solvable [11, Th. 3.6]. Although gPC expansions provide the optimal approximation to x(t) in the space of multivariate polynomials in ξ of degree ≤ p [19, Theorem 3.3], they are not of practical use for (1) because the Fourier coefficient xˆi (t) requires the knowledge of the unknown solution x(t). Then one employs the stochastic Galerkin projection technique: we approximate x(t) ≈ Pj=0 xˆ j (t)φ j (ξ ), where now xˆ j (t) are new deterministic coefficients  that depend on P and are determined by imposing Pj=0 xˆ j (t)φ j (ξ ) to be a solution to (1): ⎛ ⎞ ⎧ P P ⎪ ⎪  ⎪ ⎪ xˆ j (t)φ j (ξ ) = F ⎝t, xˆ j (t)φ j (ξ ), ζ ⎠ , ⎪ ⎨ j=0

j=0

P ⎪ ⎪ ⎪ ⎪ xˆ j (t0 )φ j (ξ ) = x0 . ⎪ ⎩ j=0

By computing the inner product with each φi (ξ ), 0 ≤ i ≤ P, and by using the orthogonality, one arrives at a deterministic system of differential equations (the Galerkin system), which may be solved via numerical techniques: ⎡ ⎛ ⎞ ⎤ ⎧ P ⎪ 1 ⎪ ⎪ ⎨xˆi (t) = E ⎣ F ⎝t, xˆ j (t)φ j (ξ ), ζ ⎠ φi (ξ )⎦ , γi j=0 ⎪ ⎪ ⎪ ⎩xˆi (t0 ) = 1 E[x0 φi (ξ )]. γi

(2)

For notational convenience, we will denote x(t) ˆ = (xˆ0 (t), . . . , xˆ P (t)) , where

is the transpose operator for vectors and matrices. Usually, Galerkin projections converge algebraically or exponentially in the mean square sense, see the numerical experiments from [2, 3, 16, 19] or the theoretical results in [12, 14]. Thus, only a small order P is required to determine reliable approximations. The expectation and the variance of x(t) are approximated as P → ∞ as follows: E[x(t)] ≈ xˆ0 (t), V[x(t)] ≈

P i=1

xˆi (t)2 γi .

(3)

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(ii) Canonical polynomial bases approach [15, 16]: This technique does not require any assumption on independence of ξ1 , . . . , ξs . We start with the canonical bases p

p

Ci = {1, ξi , . . . , ξi },

p ≥ 1, 1 ≤ i ≤ s.

We perform a simple tensor product directly, with no orthogonalisation procedure:  P = {ψ0 (ξ ), . . . , ψ P (ξ )}, where ψ0 = 1, P + 1 = ( p + s)!/( p!s!), and ψ j (ξ ) = ξ1i1 · · · ξsis , being i 1 , . . . , i s ≥ 0, i 1 + · · · + i s ≤ p and j ↔ (i 1 , . . . , i s ) in a bijective manner (for instance, via the graded lexicographic order). Note that  P is just the canonical polynomial basis of multivariate polynomials evaluated at ξ of degree ≤ p. We approximate x(t) ≈

P

x¯ j (t)ψ j (ξ ),

j=0

where x¯ j (t) are deterministic coefficients determined by imposing P

x¯ j (t)ψ j (ξ )

j=0

to be a solution to (1): ⎛ ⎞ ⎧ P P ⎪ ⎪ ⎪ ⎪ x¯ j (t)ψ j (ξ ) = F ⎝t, x¯ j (t)ψ j (ξ ), ζ ⎠ , ⎪ ⎨ j=0

j=0

P ⎪ ⎪ ⎪ ⎪ x¯ j (t0 )ψ j (ξ ) = x0 . ⎪ ⎩ j=0

By computing the inner product with each ψi (ξ ), 0 ≤ i ≤ P, we obtain the following deterministic system of differential equations: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

⎧ ⎡ ⎛ ⎞ ⎤⎫ P P ⎬ ⎨ T x¯  (t) = E ⎣ F ⎝t, x¯ j (t)ψ j (ξ ), ζ ⎠ ψi (ξ )⎦ , ⎩ ⎭ T x(t ¯ 0) =

j=0 P {E[x0 ψi (ξ )]}i=0

,

where T = (E[ψi (ξ )ψ j (ξ )])0≤i, j≤P is the Gram matrix associated to  P [20, p. 407], and x(t) ¯ = (x¯0 (t), . . . , x¯ P (t)) .

i=0

(4)

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As it has been checked numerically in [15, 16], rapid convergence of the approximations usually holds in the mean square sense, so a small P is enough. The expectation and the variance of x(t) are approximated as P → ∞ as follows: E[x(t)] ≈ V[x(t)] ≈

P

x¯ j (t)E[ψ j (ξ )], j=0 P P

(5)

x¯i (t)x¯ j (t)Cov[ψi (ξ ), ψ j (ξ )],

i=0 j=0

where Cov denotes the covariance operator. Suppose that ξ1 , . . . , ξs are independent random variables. This chapter aims at proving that the mean square approximations constructed in both methods coincide: P i=0

xˆi (t)φi (ξ ) =

P

x¯i (t)ψi (ξ ).

(6)

i=0

Hence, methodology (ii) extends the method (i) based on gPC expansions and the stochastic Galerkin projection technique. The proof of (6) will be done in the following Sect. 3. We will suppose that the deterministic systems of differential equations (2) and (4) possess the desired properties of existence and uniqueness of the solution. We would like to emphasise the fact that our reasoning is also applicable to discrete stochastic systems expressed by a random difference equation problem: 

x(m + 1) = R(m, x(m), ζ ), x(0) = x0 ,

where ζ is a random vector, x0 is a random variable, R is the deterministic recursive equation and the solution x(m) is a discrete stochastic process. In this chapter, however, we will focus on the continuous version (1). Notice that we are not dealing with stochastic systems with infinitely many random input parameters (s = ∞). For instance, we do not allow a Brownian motion in the expression of (1), or in general any stochastic process depending on infinitely many random variables. In such a situation, one performs a dimensionality reduction of the problem, for example, by truncating Karhunen-Loève expansions [19, Chap. 4]. This finite-term Karhunen-Loève expansion introduces an error, but which is usually controlled by selecting a truncation order with sufficiently small eigenvalues.

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3 Comparing gPC and Canonical Polynomial Expansions In this section, we rigorously address the proof of the identity (6). Lemma 1 The following two conditions are equivalent: 1. y(t) =

P

x¯i (t)ψi (ξ );

i=0

P 2. T x(t) ¯ = {E[y(t)ψl (ξ )]}l=0 , and E[y(t)φ j (ξ )] = 0 for each j > P.

Proof Assume condition 1. By multiplying by ψl (ξ ) and applying expectations, we obtain P . T x(t) ¯ = {E[y(t)ψl (ξ )]}l=0 On the other hand, since ψi (ξ ) is a polynomial of degree ≤ p, for 0 ≤ i ≤ P, and φ j (ξ ) is an orthogonal polynomial of degree > p, for j > P, we derive that E[ψi (ξ )φ j (ξ )] = 0. Hence, E[y(t)φ j (ξ )] =

P

x¯i (t)E[ψi (ξ )φ j (ξ )] = 0.

i=0

Assume condition 2. By the theory on convergence of gPC expansions [11] and the second part of condition 2, y(t) =

∞ E[y(t)φi (ξ )] i=0

γi

φi (ξ ) =

P E[y(t)φi (ξ )] i=0

γi

φi (ξ ).

P Hence, y(t) is a polynomial in ξ of degree ≤ p: y(t) = i=0 y¯i (t)ψi (ξ ), for certain deterministic functions y¯i (t), 0 ≤ i ≤ P. By multiplying by ψl (ξ ) and applying P . By invertibility of T and expectations, we deduce that T y¯ (t) = {E[y(t)ψl (ξ )]}l=0 the first part of condition 2, we obtain y¯ (t) = x(t), ¯ and condition 1 follows.  By applying Lemma 1 with y(t) =

P

xˆi (t)φi (ξ ),

i=0

we derive that (6) is equivalent to T x(t) ¯ = q(t), where ql (t) = E[y(t)ψl (ξ )] =

P

xˆi (t)E[φi (ξ )ψl (ξ )]

i=0 ∞ (the other condition E[y(t)φ j (ξ )] = 0, for each j > P, is clear because {φi (ξ )}i=0 is orthogonal). Let us make use of a more convenient notation. Consider the matrix

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Q with entries Q li = E[φi (ξ )ψl (ξ )], 0 ≤ l, i ≤ P. Then q(t) = Q x(t). ˆ Let S = ¯ = S x(t). ˆ T −1 Q. To sum up, we have that (6) is equivalent to x(t) ˆ By uniqueness of (4), we need to check Let z(t) = (z 0 (t), . . . , z P (t)) = S x(t). that z(t) satisfies the same differential equation problem (4) as x(t): ¯ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

   

T z (t) = E F t, T z(t0 ) =



P

z i (t)ψi (ξ ), ζ i=0  P E[x0 ψ j (ξ )] j=0 .

 P ψ j (ξ )

, j=0

(7)

The initial condition of (7) is easy to check: from the initial condition of (2), we have P  ˆ 0 ) = Q x(t ˆ 0 ) = Q E[x0 φ j (ξ )]/γ j j=0 , T z(t0 ) = T S x(t so acting componentwise for 0 ≤ i ≤ P, (T z(t0 ))i =

P

Qi j

j=0

=

P

E[x0 φ j (ξ )] γj

E[φ j (ξ )ψi (ξ )]

j=0 

= E x0

P j=0

E[x0 φ j (ξ )] γj

E[ψi (ξ )φ j (ξ )] γj



φ j (ξ )

= E[x0 ψi (ξ )], therefore the initial condition of (7) holds. To check that the differential expression from (7) holds, we need the following auxiliary lemma. Lemma 2 The matrix S has its j-th column formed by the coefficients of φ j (ξ ) in the canonical basis {ψ0 (ξ ), . . . , ψ P (ξ )}, for 0 ≤ j ≤ P: φ j (ξ ) =

P

Sk j ψk (ξ ), 0 ≤ j ≤ P.

k=0

Proof Given S defined in that way, let us see that T S = Q. We have

Is It Worthwhile Considering Orthogonality in Generalised Polynomial …

(T S)l j = =

P

269

Tlk Sk j

k=0 P

E[ψl (ξ )ψk (ξ )]Sk j   P =E Sk j ψk (ξ ) ψl (ξ ) k=0 

k=0

= E[φ j (ξ )ψl (ξ )] = Ql j . 

This completes the proof of the lemma.

From T z  (t) = T S xˆ  (t) = Q xˆ  (t) and z i (t) = (S x(t)) ˆ i , the differential expression from (7) is equivalent to    

Q xˆ (t) = E F t,

P



 P

(S x(t)) ˆ i ψi (ξ ), ζ ψ j (ξ )

.

i=0

j=0

Let us see this equality componentwise. On the one hand, (Q xˆ  (t)) j =

P

Q jl xˆl (t)

l=0

P

E[φl (ξ )ψ j (ξ )]xˆl (t) l=0   P   = E ψ j (ξ ) xˆl (t)φl (ξ ) . =

(8)

l=0

On the other hand,   P (S x(t)) ˆ E F t, i ψi (ξ ), ζ ψ j (ξ )  

 

i=0

= E F t, 

P P

 

= E ψ j (ξ )F t,



Sil xˆl (t)ψi (ξ ), ζ ψ j (ξ )

i=0 l=0

= E ψ j (ξ )F t, 



 P P l=0 P l=0





Sil ψi (ξ ) xˆl (t), ζ

i=0

xˆl (t)φl (ξ ), ζ

 ,

(9)

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where Lemma 2 has been applied in the last step. The two expressions obtained in (8) and (9) are equal, as a consequence of (2):  E ψ j (ξ )

 P

 xˆl (t)φl (ξ )





= E ψ j (ξ )F t,

P

l=0

 xˆl (t)φl (ξ ), ζ

.

l=0

Indeed, given ψ j (ξ ), we can write it as ψ j (ξ ) =

P

α jr φr (ξ )

r =0

for certain deterministic coefficients α jr , since {φ0 (ξ ), . . . , φ P (ξ )} is a basis of the multivariate polynomials evaluated in ξ of degree less than or equal to p. Then, by (2),  E ψ j (ξ )

 P

 xˆl (t)φl (ξ )

=

P r =0

l=0

=

P

α jr xˆr (t)γr   α jr E F t,

r =0





= E ψ j (ξ )F t,

P

 xˆl (t)φl (ξ ), ζ φr (ξ )

l=0 P



xˆl (t)φl (ξ ), ζ

 ,

l=0

which is the equality between (8) and (9), as wanted. We summarise this exposition in the following theorem. Theorem 3.1 Consider the approximation to x(t) constructed by means of gPC expansionsand the stochastic Galerkin projection technique with orthogonal polynoP xˆi (t)φi (ξ ). Consider the approximation to x(t) constructed directly mials (i): i=0 P from the canonical polynomial bases (ii): i=0 x¯i (t)ψi (ξ ). Then both approximations coincide: P P xˆi (t)φi (ξ ) = x¯i (t)ψi (ξ ). i=0

i=0

The conclusion is that the recent methodology (ii) extends (i). So the question is why gPC expansions and stochastic Galerkin projections based on orthogonal polynomials (method (i)) are usually employed, instead of expanding directly in terms of canonical polynomial bases. The answer to this question relies on numerical aspects. In what follows, we enumerate some advantages and disadvantages of utilising (ii) instead of the classical approach (i):

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Advantages of (ii): • Methodology (ii) works when the input random parameters are not independent. • In (ii), one does not require orthogonal polynomials. Disadvantages of (ii): • The matrix T is ill-conditioned, especially as P grows. We have checked this fact numerically. For example, if s = 1 and ξ = ξ1 ∼ Uniform (0, 1), then T is the so-called Hilbert matrix, which is the typical example of ill-conditioned matrix. Note that in procedure (i), T = Id, which is the best-conditioned matrix. • In theoretical proofs regarding the convergence of the polynomial representations, dealing with a matrix T is obviously harder than working with Id. As the convergence of the approximations is usually spectral, a small order P suffices, therefore the disadvantages of (ii) may not arise. However, if a large P is required, then disastrous numerical errors may appear. This is the reason why methodology (i) is more convenient in general. Remark 3.1 Our exposition is straightforwardly generalisable to random differential equations of higher order: ⎧ (n)  (n−1) ⎪ (t), ζ ), ⎪x (t) = F(t, x(t), x (t), . . . , x ⎪ ⎪ ⎪ ⎪ ⎨x(t0 ) = x0 , x  (t0 ) = x1 , ⎪ ⎪ ⎪ ..., ⎪ ⎪ ⎪ ⎩x (n−1) (t ) = x , 0 n−1 where ζ ∈ Rs−n is a finite random vector, x0 , x1 , . . . , xn−1 are random variables, and F is a deterministic real function F : [t0 , ∞) × Rn × Rs−n → R.

4 Numerical Experiments In this section, we illustrate the theoretical discussion of the chapter. Our objective is to emphasise the numerical issues that methodology (ii) may entail. We will work with Airy’s random differential equation: 

x  (t) + tξ x(t) = 0, t ∈ R, x(0) = x  (0) = 1,

(10)

where ξ is a random variable. This stochastic model has been previously studied in the literature with gPC expansions, see [21, 22]. It is well known that the solution to the deterministic Airy’s differential equation is highly oscillatory, hence it is expected that, in dealing with its stochastic counterpart, differences between distinct methods

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will be highlighted. Conclusions and suggestions from our exposition will be drawn in the following Sect. 5. For this problem, the systems (2) and (4) can be written as follows: 



xˆ  (t) + t Aˆ x(t) ˆ = 0, t ∈ R, x(0) ˆ = xˆ  (0) = δ0 ,

(11)

¯ = 0, t ∈ R, T x¯  (t) + t A¯ x(t)  T x(0) ¯ = T x¯ (0) = e,

(12)

where Aˆ i j = E[ξ φi (ξ )φ j (ξ )]/γi , A¯ i j = E[ξ ψi (ξ )ψ j (ξ )], ei = E[ψi (ξ )], δ0,i = 1 if i = 0 and 0 if i > 0, 0 ≤ i, j ≤ P. Observe that these systems possess the same structure as the original problem (10); this is a property of linear differential equations. The computations are done in the software Mathematica® (Wolfram Research, Inc. Mathematica, Version 12.0, Champaign, Illinois, 2019), owing to its capability to handle both numeric and symbolic computations. Example 4.1 Let us take ξ ∼ Uniform (0, 1). This distribution belongs to the Askey scheme, and Legendre polynomial orthogonal expansions can be utilised for the gPC approach. We compare the results with canonical polynomial basis expansions. We use basis degree p = 20 (so P = p = 20 because s = 1). The systems (2) and (4), which reduce to (11) and (12) in this case, have been solved numerically by running the built-in function NDSolve with no specified options. The expectations correˆ A, ¯ e and T , have been calculated sponding to the coefficients, namely the entries of A, symbolically by employing the built-in command Expectation. In Table 1 and Table 2, we report estimates of the expectation and the variance, respectively (formulas (3) and (5)), together with crude Monte Carlo simulations of order 1,000,000 to validate the results. The results are tabulated up to six significant digits. In Fig. 1 and Fig. 2, we depict the estimates of the expectation and the variance using formulas (3) and (5) and Monte Carlo simulation as t grows. The Monte Carlo simulation allows for validating the estimates obtained; although it is easy to implement and robust (it works for any large t), its convergence rate is slow. Observe that methodology (ii) does not approximate as accurately as (i) and breaks down earlier for both statistics: for t = 13 for the expectation and at t = 10 for the variance. This is due to T being the ill-conditioned Hilbert matrix of size 21 × 21. Its condition number, defined as the ratio of its maximum and minimum eigenvalue, is ≈ 1030 . Despite employing the symbolic closed-form expression of the inverse of the Hilbert matrix for T −1 , significant numerical errors appear in the end, mainly in the resolution of the differential equations system. Even the variance for t = 11 is estimated numerically as a negative number by method (ii). Example 4.2 Consider ξ ∼ Triangular(0.3, 0.7). The triangular distribution is defined via a density function with support having endpoints 0.3 and 0.7, with a

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Table 1 Approximation of E[x(t)]. Comparison between methods (i), (ii) and Monte Carlo simulation in Example 4.1 t gPC (i) p = 20 Canonical basis (ii) Monte Carlo p = 20 1,000,000 0 1 2 3 4 5 6 7 8 9 10 11 12

1 1.87749 1.85501 0.563354 −0.337611 0.208069 0.178115 −0.110055 0.187744 −0.0861767 0.125151 −0.0618336 0.0912033

1 1.87749 1.85501 0.563354 −0.337611 0.208069 0.178115 −0.110055 0.187744 −0.0861766 0.125151 −0.0618413 0.0905285

1 1.87750 1.85514 0.563561 −0.337696 0.208091 0.178893 −0.11077 0.186851 −0.0849747 0.123785 −0.0602915 0.0903416

Table 2 Approximation of V[x(t)]. Comparison between methods (i), (ii) and Monte Carlo simulation in Example 4.1 t gPC (i) p = 20 Canonical basis (ii) Monte Carlo p = 20 1,000,000 0 1 2 3 4 5 6 7 8 9 10 11 12

0 0.00490281 0.372618 2.23330 2.48375 2.31760 2.04973 1.95873 1.86612 1.81859 1.73276 1.66981 1.59450

0 0.00490281 0.372618 2.23330 2.48375 2.31760 2.04973 1.95873 1.86612 1.81944 2.19960 −535.400 −184776

0 0.00490114 0.372509 2.23313 2.48488 2.31635 2.04993 1.95917 1.86595 1.82039 1.73573 1.67041 1.59550

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Fig. 1 Approximation of E[x(t)]. Comparison between methods (i) (solid line), (ii) (dashed line) and Monte Carlo simulation (circles). This figure corresponds to Example 4.1

Fig. 2 Approximation of V[x(t)]. Comparison between methods (i) (solid line), (ii) (dashed line) and Monte Carlo simulation (circles). This figure corresponds to Example 4.1

triangular shape and with mode at 0.5. This distribution does not belong to the Askey scheme; it does not dispose of a closed family of orthogonal polynomials. Therefore a Gram-Schmidt orthonormalisation procedure is carried out. The built-in function Orthogonalise is employed, using symbolic computations (otherwise there may be a loss of orthogonality). We set p = 10 (so P = p = 10 because s = 1). Symbolic arithmetic with the built-in command Expectation has also been used ˆ for the expectations corresponding to the coefficients, namely the components of A, ¯ e and T . The condition number of T , computed as the quotient of its maximum A, and minimum eigenvalue, is ≈ 1023 . In contrast to the previous example, the inverse of T is not known in closed-form expression, so it is computed numerically using the built-in command Inverse. The systems (11) and (12) are solved numerically by means of the standard NDSolve instruction, with no added options. Figures 3 and 4 report the approximations of the expectation and the variance of x(t) as t grows, respectively, for gPC expansions, canonical polynomial expansions and crude Monte Carlo simulation. Here, the Monte Carlo simulation serves as a validation tool. For the expectation, there is good agreement between the expansions up to time 40 (they overlap in the plot), even though they deviate from the Monte Carlo estimate for t ≥ 34. However, for the variance, the canonical polynomial expansion breaks down for t ≥ 14. The gPC expansions hold more time, until t = 21. Moreover, for t > 21,

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Fig. 3 Approximation of E[x(t)]. Comparison between methods (i) (solid line), (ii) (dashed line) and Monte Carlo simulation (circles). This figure corresponds to Example 4.2

Fig. 4 Approximation of V[x(t)]. Comparison between methods (i) (solid line), (ii) (dashed line) and Monte Carlo simulation (circles). This figure corresponds to Example 4.2

albeit no good approximations are obtained, the evolution of the variance of x(t) is more or less captured.

5 Conclusions and Suggestions In this chapter, we have analysed the problem of uncertainty quantification for stochastic systems by using polynomial representations for the output stochastic process. We have proved that the approximations constructed by means of gPC expansions and the stochastic Galerkin projection technique with orthogonal polynomials are equal to the representations constructed directly from the canonical polynomial basis. Since the latter representations allow dealing with statistically dependent random input parameters, we have validated that they generalise gPC-based expansions with orthogonal polynomials. However, the Gram matrix T associated to the canonical polynomial basis is ill-conditioned, so numerical errors may arise with this new approach, and utilising orthogonal polynomials and the classical gPC method may be preferable in practice.

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We finish the exposition with some suggestions. Suppose that the random input coefficients are independent. If their distributions belong to the Askey scheme, then the best approach is that described in [1], in which one takes the associated family of orthogonal polynomials (Hermite, Legendre, etc.). In this case, no numerical errors arise a priori, as no Gram–Schmidt orthogonalisation procedure has to be carried out. If the probability distributions do not belong to the Askey scheme, one may use a Gram–Schmidt orthogonalisation procedure [9]. In general, a small order of truncation P for the Galerkin projections usually suffices due to spectral convergence, so no numerical difficulties arise. In the case that the random input coefficients are both independent and non-independent, the approach based on canonical polynomial bases [15] is appropriate, but always taking into account the fact that the Gram matrix T is ill-conditioned, especially as P grows. We would like to remark that, if the random input parameters follow a multivariate Gaussian distribution, then it is indeed possible to write them as a function of independent Normal (0, 1) random variables, and therefore Hermite polynomials may be preferable for the representations [19, Sect. 4.1.1], [22]. In practice, the best methodology is to test numerically with different techniques in order to validate each other and choose the best computational method according to the specific problem under study. Acknowledgements This work has been supported by the Spanish Ministerio de Economía, Industria y Competitividad (MINECO), the Agencia Estatal de Investigación (AEI) and Fondo Europeo de Desarrollo Regional (FEDER UE) grant MTM2017-89664-P. Conflict of Interest Statement The authors declare that there is no conflict of interests regarding the publication of this article.

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